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Solutions to problems about potentially Ks,t-bigraphic pair

  • Jian-Hua Yin EMAIL logo und Liang Zhang
Veröffentlicht/Copyright: 24. Juni 2022
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 20 Heft 1

Abstract

Let S = ( a 1 , … , a m ; b 1 , … , b n ) , where a 1 , … , a m and b 1 , … , b n are two nonincreasing sequences of nonnegative integers. The pair S = ( a 1 , … , a m ; b 1 , … , b n ) is said to be a bigraphic pair if there is a simple bipartite graph G = ( X ∪ Y , E ) such that a 1 , … , a m and b 1 , … , b n are the degrees of the vertices in X and Y , respectively. In this case, G is referred to as a realization of S . Given a bigraphic pair S , and a complete bipartite graph K s , t , we say that S is a potentially K s , t -bigraphic pair if some realization of S contains K s , t as a subgraph (with s vertices in the part of size m and t in the part of size n ). Ferrara et al. (Potentially H-bigraphic sequences, Discuss. Math. Graph Theory 29 (2009), 583–596) defined σ ( K s , t , m , n ) to be the minimum integer k such that every bigraphic pair S = ( a 1 , … , a m ; b 1 , … , b n ) with σ ( S ) = a 1 + ⋯ + a m ≥ k is a potentially K s , t -bigraphic pair. This problem can be viewed as a “potential” degree sequence relaxation of the (forcible) Turán problem. Ferrara et al. determined σ ( K s , t , m , n ) for n ≥ m ≥ 9 s 4 t 4 . In this paper, we further determine σ ( K s , t , m , n ) for n ≥ m ≥ s and n + m ≥ 2 t 2 + t + s . As two corollaries, if n ≥ m ≥ t 2 + t + s 2 or if n ≥ m ≥ s and n ≥ 2 t 2 + t , the values σ ( K s , t , m , n ) are determined completely. These results give a solution to a problem due to Ferrara et al. and a solution to a problem due to Yin and Wang.

Keywords: bigraphic pair; realization; potentially Ks,t-bigraphic pair
MSC 2010: 05C07; 05C35

1 Introduction

The study of vertex degrees in graphs has a long history, often asking when an n -tuple of nonnegative integers is realizable as the vertex degrees of a simple n -vertex graph with specified properties. Analogous problems are also studied for bipartite graphs. Let S = ( a 1 , … , a m ; b 1 , … , b n ) , where a 1 , … , a m and b 1 , … , b n are two sequences of nonnegative integers with a 1 ≥ ⋯ ≥ a m and b 1 ≥ ⋯ ≥ b n . We say that S is a bigraphic pair if there is a simple bipartite graph G with partite sets { x 1 , … , x m } and { y 1 , … , y n } such that the degree of x i is a i and the degree of y j is b j . In this case, we say that G is a realization of S . Two methods to determine if S is a bigraphic pair are the Gale-Ryser criteria [1,2] and the Havel-Hakimi-type algorithm [3].

Theorem 1.1

[1,2] S is a bigraphic pair if and only if ∑ i = 1 m a i = ∑ i = 1 n b i and ∑ i = 1 k a i ≤ ∑ i = 1 n min { k , b i } for k = 1 , … , m (or ∑ i = 1 k b i ≤ ∑ i = 1 m min { k , a i } for k = 1 , … , n ).

For 1 ≤ p ≤ m and 1 ≤ q ≤ n , let S ( a p ) = ( a 1 , … , a p − 1 , a p + 1 , … , a m ; b 1 ′ , … , b n ′ ) and S ( b q ) = ( a 1 ′ , … , a m ′ ; b 1 , … , b q − 1 , b q + 1 , … , b n ) , where b 1 ′ ≥ ⋯ ≥ b n ′ is a rearrangement in nonincreasing order of b 1 − 1 , … , b a p − 1 , b a p + 1 , … , b n and a 1 ′ ≥ ⋯ ≥ a m ′ is a rearrangement in nonincreasing order of a 1 − 1 , … , a b q − 1 , a b q + 1 , … , a m . We say that S ( a p ) (resp. S ( b q ) ) is the residual pair obtained from S by laying off a p (resp. b q ).

Theorem 1.2

[3] S is a bigraphic pair if and only if S ( a p ) (or S ( b q ) ) is a bigraphic pair.

We can also ask whether there is a realization satisfying a particular property. Let S = ( a 1 , … , a m ; b 1 , … , b n ) be a bigraphic pair, and let K s , t be the complete bipartite graph with partite sets of size s and t . We say that S is a potentially K s , t -bigraphic pair if some realization of S contains K s , t (with s vertices in the part of size m and t in the part of size n ). If some realization of S contains K s , t on those vertices having degree a 1 , … , a s , b 1 , … , b t , we say that S is a potentially A s , t -bigraphic pair. Ferrara et al. [4] proved that S is a potentially A s , t -bigraphic pair if and only if it is a potentially K s , t -bigraphic pair. Yin and Wang [5] developed a Havel-Hakimi-type algorithm to determine if S is a potentially K s , t -bigraphic pair. This algorithm can also be used to construct a graph with degree sequence pair S and containing K s , t on those vertices having degree a 1 , … , a s , b 1 , … , b t .

Let S = ( a 1 , … , a m ; b 1 , … , b n ) , where a 1 , … , a m and b 1 , … , b n are two nonincreasing sequences of nonnegative integers. Let 1 ≤ s ≤ m , 1 ≤ t ≤ n , a s ≥ t and b t ≥ s . We first define pairs S 0 , … , S s as follows. Let S 0 = S . Let

S 1 = ( a 2 , … , a m ; b 1 − 1 , … , b t − 1 , b t + 1 ( 1 ) , … , b n ( 1 ) ) ,

where b t + 1 ( 1 ) ≥ ⋯ ≥ b n ( 1 ) is a rearrangement in nonincreasing order of b t + 1 − 1 , … , b a 1 − 1 , b a 1 + 1 , … , b n . For 2 ≤ i ≤ s , given S i − 1 = ( a i , … , a m ; b 1 − i + 1 , … , b t − i + 1 , b t + 1 ( i − 1 ) , … , b n ( i − 1 ) ) , let

S i = ( a i + 1 , … , a m ; b 1 − i , … , b t − i , b t + 1 ( i ) , … , b n ( i ) ) ,

where b t + 1 ( i ) ≥ ⋯ ≥ b n ( i ) is a rearrangement in nonincreasing order of b t + 1 ( i − 1 ) − 1 , … , b a i ( i − 1 ) − 1 , b a i + 1 ( i − 1 ) , … , b n ( i − 1 ) .

We now define pairs S 0 ′ , … , S t ′ as follows. Let S 0 ′ = S . Let

S 1 ′ = ( a 1 − 1 , … , a s − 1 , a s + 1 ( 1 ) , … , a m ( 1 ) ; b 2 , … , b n ) ,

where a s + 1 ( 1 ) ≥ ⋯ ≥ a m ( 1 ) is a rearrangement in nonincreasing order of a s + 1 − 1 , … , a b 1 − 1 , a b 1 + 1 , … , a m . For 2 ≤ i ≤ t , given S i − 1 ′ = ( a 1 − i + 1 , … , a s − i + 1 , a s + 1 ( i − 1 ) , … , a m ( i − 1 ) ; b i , … , b n ) , let

S i ′ = ( a 1 − i , … , a s − i , a s + 1 ( i ) , … , a m ( i ) ; b i + 1 , … , b n ) ,

where a s + 1 ( i ) ≥ ⋯ ≥ a m ( i ) is a rearrangement in nonincreasing order of a s + 1 ( i − 1 ) − 1 , … , a b i ( i − 1 ) − 1 , a b i + 1 ( i − 1 ) , … , a m ( i − 1 ) .

Theorem 1.3

[5] S is a potentially A s , t -bigraphic pair if and only if S s (or S t ′ ) is a bigraphic pair.

Motivated by the problem due to Erdős et al. [6] of finding the minimum integer k such that every realizable n -tuple with a sum of at least k is potentially K r -graphic, Ferrara et al. [4] investigated analogous problem for bipartite graphs. They defined σ ( K s , t , m , n ) to be the minimum integer k such that every bigraphic pair S = ( a 1 , … , a m ; b 1 , … , b n ) with σ ( S ) = a 1 + ⋯ + a m ≥ k is a potentially K s , t -bigraphic pair. They determined σ ( K s , t , m , n ) when m and n are sufficiently large in terms of s and t . This problem can be viewed as a “potential” degree sequence relaxation of the (forcible) Turán problem.

Theorem 1.4

[4] If t ≥ s ≥ 1 and n ≥ m ≥ 9 s 4 t 4 , then σ ( K s , t , m , n ) = n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 .

Ferrara et al. proposed a problem as follows.

Problem 1.1

[4] This would be useful if one were interested in finding smaller bounds on the n and m necessary to assure Theorem 1.4.

Yin and Wang proved a new result as follows.

Theorem 1.5

[5] If t ≥ s ≥ 1 , n ≥ m ≥ s and n ≥ ( s + 1 ) t 2 − ( 2 s − 1 ) t + s − 1 , then σ ( K s , t , m , n ) = n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 .

Yin and Wang also proposed a problem as follows.

Problem 1.2

[5] It would be meaningful to investigate a lower bound on n + m necessary to assure Theorem 1.5.

The purpose of this paper is to improve Theorem 1.5 and determine σ ( K s , t , m , n ) for n ≥ m ≥ s and n + m ≥ 2 t 2 + t + s , that is, a solution to Problems 1.2. As two corollaries, if n ≥ m ≥ t 2 + t + s 2 or if n ≥ m ≥ s and n ≥ 2 t 2 + t , the values σ ( K s , t , m , n ) are determined completely, that is, a solution to Problem 1.1.

Theorem 1.6

If t ≥ s ≥ 1 , n ≥ m ≥ s and n + m ≥ 2 t 2 + t + s , then σ ( K s , t , m , n ) = n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 .

Corollary 1.1

If t ≥ s ≥ 1 and n ≥ m ≥ t 2 + t + s 2 , then σ ( K s , t , m , n ) = n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 .

Corollary 1.2

If t ≥ s ≥ 1 , n ≥ m ≥ s and n ≥ 2 t 2 + t , then σ ( K s , t , m , n ) = n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 .

2 Proof of Theorem 1.6

In order to prove Theorem 1.6, we need some lemmas.

Lemma 2.1

[7] Theorem 1.1 remains valid if ∑ i = 1 k a i ≤ ∑ i = 1 n min { k , b i } is assumed only for those k for which a k > a k + 1 or k = m (or ∑ i = 1 k b i ≤ ∑ i = 1 m min { k , a i } is assumed only for those k for which b k > b k + 1 or k = n ).

Lemma 2.2

[5] Let S = ( a 1 , … , a m ; b 1 , … , b n ) be a bigraphic pair with a s ≥ t , b t ≥ s , m − 1 ≥ b 1 ≥ ⋯ ≥ b t = ⋯ = b a 1 + 1 ≥ b a 1 + 2 ≥ ⋯ ≥ b n and n − 1 ≥ a 1 ≥ ⋯ ≥ a s = ⋯ = a b 1 + 1 ≥ a b 1 + 2 ≥ ⋯ ≥ a m . For each S i = ( a i + 1 , … , a m ; b 1 − i , … , b t − i , b t + 1 ( i ) , … , b n ( i ) ) with 0 ≤ i ≤ s , let t i = max { j ∣ b t + 1 ( i ) − b t + j ( i ) ≤ 1 } . Then

  1. t s ≥ t s − 1 ≥ ⋯ ≥ t 0 ≥ a 1 + 1 − t .

  2. For each i with 1 ≤ i ≤ s , we have b t + k ( i ) = b t + k ( i − 1 ) for k > t i . Consequently, b t + k ( s ) = b t + k for k > t s .

Lemma 2.3

Let S = ( a 1 , … , a m ; b 1 , … , b n ) be a bigraphic pair with a s ≥ t , b t ≥ s , m − 1 ≥ b 1 ≥ ⋯ ≥ b t = ⋯ = b a 1 + 1 ≥ b a 1 + 2 ≥ ⋯ ≥ b n and n − 1 ≥ a 1 ≥ ⋯ ≥ a s = ⋯ = a b 1 + 1 ≥ a b 1 + 2 ≥ ⋯ ≥ a m . If ∑ i = 1 t ( b i − b t ) + ∑ i = a s + 1 + 1 n b i ≥ t s , then S is a potentially A s , t -bigraphic pair.

Proof

It is trivial for s = 1 . Assume s ≥ 2 . By Theorem 1.3, we only need to check that S s = ( a s + 1 , … , a m ; b 1 − s , … , b t − s , b t + 1 ( s ) , … , b n ( s ) ) is a bigraphic pair. Clearly, a s + 1 + ⋯ + a m = ( b 1 − s ) + ⋯ + ( b t − s ) + b t + 1 ( s ) + ⋯ + b n ( s ) . Denote ℓ = b t and p = max { i ∣ a s + i = a s } . Then s + p ≥ b 1 + 1 , i.e., p ≥ b 1 + 1 − s . By Lemma 2.1, it is enough to check that ∑ i = 1 k a s + i ≤ ∑ i = 1 t min { k , b i − s } + ∑ i = t + 1 n min { k , b i ( s ) } for p ≤ k ≤ m − s . Denote x = b t + 1 ( s ) . By b t + 1 ( s ) ≤ b t + 1 = ℓ , we have x ≤ ℓ . If k ≥ x , by k ≥ p ≥ b 1 + 1 − s > b i − s for 1 ≤ i ≤ t , then ∑ i = 1 t min { k , b i − s } + ∑ i = t + 1 n min { k , b i ( s ) } = ∑ i = 1 t ( b i − s ) + ∑ i = t + 1 n b i ( s ) = a s + 1 + ⋯ + a m ≥ ∑ i = 1 k a s + i . Assume p ≤ k ≤ x − 1 . If t s ≥ a s + 1 , by b a s + 1 + t ≥ b t + t s ≥ x − 1 ≥ k , then ∑ i = 1 t min { k , b i − s } + ∑ i = t + 1 n min { k , b i ( s ) } ≥ ∑ i = t + 1 a s + 1 + t min { k , b i ( s ) } = k a s + 1 ≥ ∑ i = 1 k a s + i . Assume t s < a s + 1 . Then by Lemma 2.2, b t + j ( s ) = b t + j for j ≥ a s + 1 . If k ≤ b a s + 1 + t , then ∑ i = 1 t min { k , b i − s } + ∑ i = t + 1 n min { k , b i ( s ) } ≥ ∑ i = t + 1 a s + 1 + t min { k , b i ( s ) } = k a s + 1 ≥ ∑ i = 1 k a s + i . Assume k > b a s + 1 + t . For each i with a s + 1 + 1 ≤ i ≤ t + t s , we have min { k , b i ( s ) } = k = ℓ − ( ℓ − k ) ≥ b i − ( ℓ − k ) . Also, for each i with t + t s + 1 ≤ i ≤ a s + 1 + t , by Lemma 2.2, we have min { k , b i ( s ) } = min { k , b i } = min { ℓ − ( ℓ − k ) , b i } ≥ min { b i − ( ℓ − k ) , b i } = b i − ( ℓ − k ) . Therefore, ∑ i = 1 t min { k , b i − s } + ∑ i = t + 1 n min { k , b i ( s ) } = ∑ i = 1 t ( b i − s ) + ∑ i = t + 1 a s + 1 min { k , b i ( s ) } + ∑ i = a s + 1 + 1 a s + 1 + t min { k , b i ( s ) } + ∑ i = a s + 1 + t + 1 n min { k , b i ( s ) } ≥ ∑ i = 1 t ( ( b i − ℓ ) + ( ℓ − s ) ) + k ( a s + 1 − t ) + ∑ i = a s + 1 + 1 a s + 1 + t ( b i − ( ℓ − k ) ) + ∑ i = a s + 1 + t + 1 n b i = ∑ i = 1 t ( b i − ℓ ) + ∑ i = a s + 1 + 1 n b i + ( ℓ − s ) t + k ( a s + 1 − t ) − ( ℓ − k ) t ≥ t s + ( ℓ − s ) t + k ( a s + 1 − t ) − ( ℓ − k ) t = k a s + 1 ≥ ∑ i = 1 k a s + i .□

Lemma 2.4

Let S = ( a 1 , … , a m ; b 1 , … , b n ) be a bigraphic pair with a s ≥ t , b t ≥ s , m − 1 ≥ b 1 ≥ ⋯ ≥ b t = ⋯ = b a 1 + 1 ≥ b a 1 + 2 ≥ ⋯ ≥ b n and n − 1 ≥ a 1 ≥ ⋯ ≥ a s = ⋯ = a b 1 + 1 ≥ a b 1 + 2 ≥ ⋯ ≥ a m . If ∑ i = 1 s ( a i − a s ) + ∑ i = b t + 1 + 1 m a i ≥ t s , then S is a potentially A s , t -bigraphic pair.

Proof

By the symmetry, the proof of Lemma 2.4 is similar to that of Lemma 2.3.□

Lemma 2.5

[4] Suppose that S = ( a 1 , … , a m ; b 1 , … , b n ) is not a potentially A s , t -bigraphic pair. Let G be a realization of S with partite sets X and Y , with ∣ X ∣ = m and ∣ Y ∣ = n . Let X s be the set of s highest degree vertices of X , and Y t be the set of t highest degree vertices of Y . Assume that G is a realization of S that maximizes the number of edges between X s and Y t . Let x and y be nonadjacent members of X s and Y t , and let A = N G ( x ) ⧹ Y t and B = N G ( y ) ⧹ X s . Then both A and B contain at most ( s − 1 ) ( t − 1 ) vertices.

Lemma 2.6

[7] If S = ( a 1 , … , a m ; b 1 , … , b n ) is a bigraphic pair with a s ≥ 2 t − 1 and b t ≥ 2 s − 1 , then S is a potentially A s , t -bigraphic pair.

Lemma 2.7

Let S = ( a 1 , … , a m ; b 1 , … , b n ) be a bigraphic pair with m − 1 ≥ b 1 ≥ ⋯ ≥ b t = ⋯ = b a 1 + 1 ≥ b a 1 + 2 ≥ ⋯ ≥ b n and n − 1 ≥ a 1 ≥ ⋯ ≥ a s = ⋯ = a b 1 + 1 ≥ a b 1 + 2 ≥ ⋯ ≥ a m . If n ( s − 1 ) + m ( t − 1 ) ≥ max { 2 s t 2 − 2 t 2 + t − s , 2 t s 2 − 2 s 2 + s − t } and σ ( S ) ≥ n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 , then S is a potentially A s , t -bigraphic pair.

Proof

By σ ( S ) ≥ n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 , it is straightforward to show that a s ≥ t and b t ≥ s . On the contrary, we assume that S is not a potentially A s , t -bigraphic pair. Let G be a realization of S with partite sets X and Y , with ∣ X ∣ = m and ∣ Y ∣ = n . Let X s be the set of s highest degree vertices of X , and Y t be the set of t highest degree vertices of Y . Assume that G is a realization of S that maximizes the number of edges between X s and Y t . Let x and y be nonadjacent members of X s and Y t , and let A = N G ( x ) ⧹ Y t and B = N G ( y ) ⧹ X s . By Lemma 2.5, both A and B contain at most ( s − 1 ) ( t − 1 ) vertices. This implies a s ≤ d G ( x ) ≤ ∣ A ∣ + ∣ Y t ∣ − 1 ≤ ( s − 1 ) ( t − 1 ) + t − 1 = s t − s and b t ≤ d G ( y ) ≤ ∣ B ∣ + ∣ X s ∣ − 1 ≤ ( s − 1 ) ( t − 1 ) + s − 1 = s t − t . By Lemma 2.6, we have a s ≤ 2 t − 2 or b t ≤ 2 s − 2 , and so we may consider the following two cases.

Case 1. a s ≤ 2 t − 2 .

It follows from Lemma 2.3 that σ ( S ) = ∑ i = 1 t b i + ∑ i = t + 1 a s + 1 b i + ∑ i = a s + 1 + 1 n b i = ∑ i = 1 t ( b i − b t + b t ) + ∑ i = t + 1 a s + 1 b i +  ∑ i = a s + 1 + 1 n b i ≤ ∑ i = 1 t ( b i − b t ) + ∑ i = a s + 1 + 1 n b i  +  t b t + ( a s + 1 − t ) b t ≤ t s − 1 + a s + 1 b t ≤ t s − 1 + ( 2 t − 2 ) ( s t − t ) < ( 2 s t 2 − 2 t 2 + t − s ) − ( t − 1 ) ( s − 1 ) + 1 ≤ n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 , a contradiction.

Case 2. b t ≤ 2 s − 2 .

It follows from Lemma 2.4 that σ ( S ) = ∑ i = 1 s a i + ∑ i = s + 1 b t + 1 a i + ∑ i = b t + 1 + 1 m a i = ∑ i = 1 s ( a i − a s + a s ) + ∑ i = s + 1 b t + 1 a i + ∑ i = b t + 1 + 1 m a i ≤ ∑ i = 1 s ( a i − a s ) + ∑ i = b t + 1 + 1 m a i + s a s + ( b t + 1 − s ) a s ≤ t s − 1 + b t + 1 a s ≤ t s − 1 + ( 2 s − 2 ) ( s t − s ) < ( 2 t s 2 − 2 s 2 + s − t ) − ( t − 1 ) ( s − 1 ) + 1 ≤ n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 , a contradiction.□

Lemma 2.8

Let S = ( a 1 , … , a m ; b 1 , … , b n ) be a bigraphic pair. If n + m ≥ 2 max { t 2 , s 2 } + t + s and σ ( S ) ≥ n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 , then S is a potentially A s , t -bigraphic pair.

Proof

It is straightforward to show that a s ≥ t and b t ≥ s . We use induction on s + t . It is trivial for s = 1 or t = 1 . Assume s ≥ 2 and t ≥ 2 . If a 1 = n or there exists an integer k with t ≤ k ≤ a 1 such that b k > b k + 1 , then the residual pair S ( a 1 ) = ( a 2 , … , a m ; b 1 ′ , … , b n ′ ) obtained from S by laying off a 1 satisfies n + ( m − 1 ) ≥ 2 max { t 2 , s 2 } + t + ( s − 1 ) ≥ 2 max { t 2 , ( s − 1 ) 2 } + t + ( s − 1 ) , σ ( S ( a 1 ) ) = σ ( S ) − a 1 ≥ n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 − n = n ( s − 2 ) + ( m − 1 ) ( t − 1 ) − ( t − 1 ) ( s − 2 ) + 1 and b 1 ′ = b 1 − 1 , … , b t ′ = b t − 1 . By Theorem 1.2 and the induction hypothesis, S ( a 1 ) is a potentially A s − 1 , t -bigraphic pair, and hence S is a potentially A s , t -bigraphic pair. So we may assume a 1 ≤ n − 1 and b 1 ≥ ⋯ ≥ b t = ⋯ = b a 1 + 1 ≥ b a 1 + 2 ≥ ⋯ ≥ b n . If b 1 = m or there exists an integer k with s ≤ k ≤ b 1 such that a k > a k + 1 , then the residual pair S ( b 1 ) = ( a 1 ′ , … , a m ′ ; b 2 , … , b n ) obtained from S by laying off b 1 satisfies ( n − 1 ) + m ≥ 2 max { t 2 , s 2 } + ( t − 1 ) + s ≥ 2 max { ( t − 1 ) 2 , s 2 } + ( t − 1 ) + s , σ ( S ( b 1 ) ) = σ ( S ) − b 1 ≥ n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 − m = ( n − 1 ) ( s − 1 ) + m ( t − 2 ) − ( t − 2 ) ( s − 1 ) + 1 and a 1 ′ = a 1 − 1 , … , a s ′ = a s − 1 . By Theorem 1.2 and the induction hypothesis, S ( b 1 ) is a potentially A s , t − 1 -bigraphic pair, and hence S is a potentially A s , t -bigraphic pair. So we may further assume b 1 ≤ m − 1 and a 1 ≥ ⋯ ≥ a s = ⋯ = a b 1 + 1 ≥ a b 1 + 2 ≥ ⋯ ≥ a m . If s ≤ t , then 2 s t 2 − 2 t 2 + t − s ≥ 2 t s 2 − 2 s 2 + s − t and n ( s − 1 ) + m ( t − 1 ) ≥ ( n + m ) ( s − 1 ) ≥ ( 2 t 2 + t + s ) ( s − 1 ) = 2 s t 2 − 2 t 2 + ( t + s ) ( s − 1 ) ≥ 2 s t 2 − 2 t 2 + t + s , implying that n ( s − 1 ) + m ( t − 1 ) ≥ max { 2 s t 2 − 2 t 2 + t − s , 2 t s 2 − 2 s 2 + s − t } . Similarly, if t ≤ s , then 2 t s 2 − 2 s 2 + s − t ≥ 2 s t 2 − 2 t 2 + t − s and n ( s − 1 ) + m ( t − 1 ) ≥ ( n + m ) ( t − 1 ) ≥ ( 2 s 2 + t + s ) ( t − 1 ) = 2 t s 2 − 2 s 2 + ( t + s ) ( t − 1 ) ≥ 2 t s 2 − 2 s 2 + t + s , implying that n ( s − 1 ) + m ( t − 1 ) ≥ max { 2 s t 2 − 2 t 2 + t − s , 2 t s 2 − 2 s 2 + s − t } . Thus by Lemma 2.7, S is a potentially A s , t -bigraphic pair.□

Proof of Theorem 1.6

To show the lower bound, Ferrara et al. [4] considered the bigraphic pair S = ( n s − 1 , ( t − 1 ) m − s + 1 ; m s − 1 , ( t − 1 ) m − s + 1 , ( s − 1 ) n − m ) , where the symbol x y stands for y consecutive terms, each equal to x . Clearly, neither partite set in any realization of S has s vertices of degree t . Hence, S is not a potentially K s , t -bigraphic pair. Thus, σ ( K s , t , m , n ) ≥ σ ( S ) + 1 = n ( s − 1 ) + m ( t − 1 ) − ( t − 1 ) ( s − 1 ) + 1 . The upper bound directly follows from Lemma 2.8.□

Acknowledgements

The authors would like to thank the referees for their helpful suggestions.

  1. Funding information: This research was supported by Hainan Provincial Natural Science Foundation of China (Nos. 122RC545, 2019RC085), National Natural Science Foundation of China (No. 11961019), and Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province.

  2. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-06-26
Accepted: 2022-01-12
Published Online: 2022-06-24

© 2022 Jian-Hua Yin and Liang Zhang, published by De Gruyter

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  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0022
Schlagwörter für diesen Artikel
bigraphic pair; realization; potentially Ks,t-bigraphic pair
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Artikel in diesem Heft

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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