On the existence of tripartite graphs and n-partite graphs

Jiyun Guo; Haiyan Li; Yuqin Zhang; Lingling Liu; Miao Fu

doi:10.1515/math-2024-0093

Article Open Access

On the existence of tripartite graphs and n-partite graphs

Jiyun Guo , Haiyan Li , Yuqin Zhang , Lingling Liu and Miao Fu

Published/Copyright: December 10, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

A sequence α of nonnegative integers is said to be graphic if it is the degree sequence of a simple graph G , and such a graph G is called a realization of α . In this article, we generalize Gale and Ryser’s theorem and give the sufficient condition and necessary condition for a triple to be realized by a tripartite graph. Not only that, we also give another stronger monotonous degree condition.

Keywords: tripartite graphs; monotonous degree condition; n-partite graphs

MSC 2010: 05C07

1 Introduction

A sequence α of nonnegative integers is said to be graphic if it is the degree sequence of a simple graph G , and such a graph G is called a realization of α . Many researchers devote themselves to the study of degree sequence, such as Erdős and Gallai [1], Gale [2], and Ryser [3], among which Erdős and Gallai [1] presented the most classic characterization. Before presenting it, it is necessary to point out that the elements of sequences and interval’s endpoints covered in this article are composed of nonnegative integers.

Theorem 1.1

(Erdős and Gallai [1]) Let α = ( α 1 , α 2 , … , α m ) be a sequence with α 1 ≥ α 2 ≥ … ≥ α m and ∑ i = 1 m α i ≡ 0 ( mod 2 ) . Then α is realized by a graph if and only if

∑ i = 1 t α i ≤ t ( t − 1 ) + ∑ i = t + 1 m min { α i , t } f o r e a c h t w i t h 1 ≤ t ≤ m .

There are many ways to prove Theorem 1.1 [4–7]. In addition, the results on degree sequences in [8–10] are also worth reading. It is natural to generalize Theorem 1.1 to bipartite graphs. Gale [2] and Ryser [3], established an appreciable result of two sequences that realize a simple bipartite graph.

Theorem 1.2

(Gale [2], Ryser [3]). Let α = ( α 1 , α 2 , … , α m ) and β = ( β 1 , β 2 , … , β n ) be two sequences with α 1 ≥ α 2 ≥ … ≥ α m , β 1 ≥ β 2 ≥ … ≥ β n and ∑ i = 1 m α i = ∑ j = 1 n β j . Then ( α ; β ) is realized by a simple graph if and only if

∑ j = 1 t β j ≤ ∑ i = 1 m min { α i , t } f o r e a c h t w i t h 1 ≤ t ≤ n .

Worth mentioning that most of the results about degree sequences can be generalized to hypergraphs, and of course the results are no exception. One can refer to [11–14] for articles on hypergraphs.

In light of Theorems 1.1 and 1.2, one cannot help wondering about the characterization of tripartite graphic sequences, and motivated by the constructive extensions on graphic sequences due to Tripathi et al. [15], we give some extensions on tripartite graphic sequences.

2 Tripartite graphs

Let η 1 = ( [ a 1 , b 1 ] , … , [ a m , b m ] ) , η 2 = ( [ c 1 , d 1 ] , … , [ c n , d n ] ) , and η 3 = ( [ g 1 , h 1 ] , … , [ g l , h l ] ) be three sequences of intervals with a 1 ≥ a 2 ≥ … ≥ a m , c 1 ≥ c 2 ≥ … ≥ c n and g 1 ≥ g 2 ≥ … ≥ g l . We say that ( η 1 ; η 2 ; η 3 ) is realized by a tripartite graph (i.e., tripartite graphic in intervals) if there exists a tripartite graph G with vertex sets X = { x 1 , x 2 , … , x m } , Y = { y 1 , y 2 , … , y n } , and Z = { z 1 , z 2 , … , z l } so that a i ≤ d G ( x i ) ≤ b i for i = 1 , 2 , … , m , c j ≤ d G ( y j ) ≤ d j for j = 1 , 2 , … , n and g k ≤ d G ( z k ) ≤ h k for k = 1 , 2 , … , l . In this section, we first prove the monotonous sufficient condition for the existence of tripartite graph (i.e., 3-partite graph) by constructive approach and then prove the necessary condition for the existence of 3-partite graph.

Theorem 2.1

Let η 1 = ( [ a 1 , b 1 ] , … , [ a m , b m ] ) , η 2 = ( [ c 1 , d 1 ] , … , [ c n , d n ] ) , and η 3 = ( [ g 1 , h 1 ] , … , [ g l , h l ] ) be three sequences of intervals with a 1 ≥ a 2 ≥ … ≥ a m , c 1 ≥ c 2 ≥ … ≥ c n , and g 1 ≥ g 2 ≥ … ≥ g l . If η 1 , η 2 , and η 3 satisfy each of the following:

(1) ∑ i = 1 t a i 2 ≤ ∑ j = 1 n min d j 2 , t f o r e v e r y t w i t h 1 ≤ t ≤ m ,

(2) ∑ i = 1 t a i 2 ≤ ∑ j = 1 l min h j 2 , t f o r e v e r y t w i t h 1 ≤ t ≤ m ,

(3) ∑ i = 1 p c i 2 ≤ ∑ j = 1 m min b j 2 , p f o r e v e r y p w i t h 1 ≤ p ≤ n ,

(4) ∑ i = 1 p c i 2 ≤ ∑ j = 1 l min h j 2 , p f o r e v e r y p w i t h 1 ≤ p ≤ n ,

(5) ∑ i = 1 q g i 2 ≤ ∑ j = 1 m min b j 2 , q f o r e v e r y q w i t h 1 ≤ q ≤ l ,

(6) ∑ i = 1 q g i 2 ≤ ∑ j = 1 n min d j 2 , q f o r e v e r y q w i t h 1 ≤ q ≤ l ,

then ( η 1 ; η 2 ; η 3 ) is realized by a 3-partite graph.

Proof

Tripathi et al. [15] developed two results on graphic sequences using a constructive approach. To prove the conclusion, we refer to their algorithm. Now consider three sequences of intervals η 1 = ( [ a 1 , b 1 ] , … , [ a m , b m ] ) , η 2 = ( [ c 1 , d 1 ] , … , [ c n , d n ] ) and η 3 = ( [ g 1 , h 1 ] , … , [ g l , h l ] ) with a 1 ≥ a 2 ≥ … ≥ a m , c 1 ≥ c 2 ≥ … ≥ c n and g 1 ≥ g 2 ≥ … ≥ g l . Assume that η 1 , η 2 , η 3 satisfy the conditions (1)–(6). Our goal is to construct a 3-partite graph G , which consists of three bipartite graphs G 1 , G 2 , and G 3 (i.e., G = G 1 ∪ G 2 ∪ G 3 ) with bipartition ( X , Y ) , ( X , Z ) and ( Y , Z ) respectively. In fact, the existence of each graph G 1 , G 2 , and G 3 is assured by Theorem 1.2. Next, constructive method is applied to reconstruct G 1 , G 2 , and G 3 .

To construct the graph G 1 , we first construct a bigraph G 1 ′ with partition X = { x 1 , x 2 , … , x m } and Y = { y 1 , y 2 , … , y n } such that d Y ( x i ) = a i 2 for i = 1 , 2 , … , m and d X ( y j ) ≤ d j 2 for j = 1 , 2 , … , n , where d Y ( x i ) denotes the contribution to the degree of vertex x i from edges incident to vertices in Y . Call a graph G 0 a sub-realization if d Y ( x i ) ≤ b i 2 for i = 1 , 2 , … , m and d X ( y j ) ≤ d j 2 for j = 1 , 2 , … , n , and a realization if a i 2 ≤ d Y ( x i ) ≤ b i 2 for all i , and c j 2 ≤ d X ( y j ) ≤ d j 2 for all j . In a subrealization, let the critical value r be the maximum index for which d Y ( x i ) = a i 2 for 1 ≤ i < r and d Y ( x r ) < a r 2 hold. Initially, we may start with an empty graph with vertex set X ∪ Y , so that r = 1 unless a i = 0 for all i , in which case the process is complete. While r ≤ m , we obtain a new sub-realization by iteratively removing the deficiency a r 2 − d Y ( x r ) at vertex x r while maintaining d Y ( x i ) = a i 2 for 1 ≤ i < r and d X ( y j ) ≤ d j 2 for 1 ≤ j ≤ n .

There must be a vertex v ∈ N Y ( x i ) \ N Y ( x r ) for 1 ≤ i < r , since d Y ( x i ) = a i 2 ≥ a r 2 > d Y ( x r ) , where N Y ( x i ) denotes the neighbor set of x i in Y . We write x i ↔ y j for “ x i is adjacent to y j ” and x i ↮ y j for “ x i is not adjacent to y j ”.

Case 1. Suppose for some j , k > r and w ≤ r , y j ↔ x k and y j ↮ x w . If w = r , replace x k y j by x r y j . If w < r , replace x k y j , x w v by x w y j , x r v , where v ∈ N Y ( x w ) \ N Y ( x r ) .

Case 2. Suppose for some j and w ≤ r , d X ( y j ) < d j 2 and y j ↮ x w . If w = r , connect vertices x r and y j . If w < r , replace x w v by x w y j , x r v , where v ∈ N Y ( x w ) \ N Y ( x r ) .

If neither of the cases is running anymore, then

(7) ∑ i = 1 r − 1 a i 2 + d Y ( x r ) = ∑ i = 1 r d Y ( x i ) = ∑ j = 1 n min { d X ( y j ) , r } = ∑ j = 1 n min d j 2 , r .

By (1), (7) and d Y ( x r ) ≤ a r 2 , we have d Y ( x r ) = a r 2 . Increasing the value of r by one, and applying similar steps yields the bipartite graphs G 1 ′ with vertex sets X = { x 1 , x 2 , … , x m } and Y = { y 1 , y 2 , … , y n } satisfying d Y ( x i ) = a i 2 for i = 1 , 2 , … , m and d X ( y j ) ≤ d j 2 for j = 1 , 2 , … , n .

Next, we shall construct a new bipartite graph G 1 based on G 1 ′ . Let the critical value s be the maximum subscript for which d X ( y j ) ≥ c j 2 for all j < s and d X ( y s ) < c s 2 hold. We will remove the difference at vertex y s while maintaining d X ( y j ) ≥ c j 2 for all j < s and a i 2 ≤ d Y ( x i ) ≤ b i 2 for all i ≤ m unless another situation arises.

Case 3. Suppose d X ( y j ) > c j 2 for 1 ≤ j < s , then replace y j v by y s v , where v ∈ N X ( y j ) \ N X ( y s ) .

If Cases 1–3 are no longer executed, then analogous to (7), we obtain d X ( y s ) = c s 2 . Adding 1 to the value of s , and continuing similar steps yields the required bipartite graph G 1 with vertex sets X = { x 1 , x 2 , … , x m } and Y = { y 1 , y 2 , … , y n } so that a i 2 ≤ d Y ( x i ) ≤ b i 2 for i = 1 , 2 , … , m and d X ( y j ) = c j 2 for j = 1 , 2 , … , n . Similarly, we can obtain G 2 with vertex sets X = { x 1 , x 2 , … , x m } and Z = { z 1 , z 2 , … , z l } so that a i 2 ≤ d Z ( x i ) ≤ b i 2 for i = 1 , 2 , … , m and d X ( z k ) = g k 2 for k = 1 , 2 , … , l . We can also obtain G 3 with vertex sets Y = { y 1 , y 2 , … , y n } and Z = { z 1 , z 2 , … , z l } so that c j 2 ≤ d Z ( y j ) ≤ d j 2 for j = 1 , 2 , … , n and d Y ( z k ) = g k 2 for k = 1 , 2 , … , l . In conclusion, we obtain three bipartite graphs called G 1 , G 2 , and G 3 , where

G 1 : a i 2 ≤ d Y ( x i ) ≤ b i 2 for 1 ≤ i ≤ m , d X ( y j ) = c j 2 for 1 ≤ j ≤ n ; G 2 : a i 2 ≤ d Z ( x i ) ≤ b i 2 for 1 ≤ i ≤ m , d X ( z k ) = g k 2 for 1 ≤ k ≤ l ; G 3 : c j 2 ≤ d Z ( y j ) ≤ d j 2 for 1 ≤ j ≤ n , d Y ( z k ) = g k 2 for 1 ≤ k ≤ l .

Set G = G 1 ∪ G 2 ∪ G 3 , then G is the required 3-partite graph, since d G ( x i ) = d Y ( x i ) + d Z ( x i ) , d G ( y j ) = d X ( y j ) + d Z ( y j ) , and d G ( z k ) = d X ( z k ) + d Y ( z k ) . Clearly, 2 a i 2 ≤ d G ( x i ) ≤ 2 b i 2 for 1 ≤ i ≤ m , 2 c j 2 ≤ d G ( y j ) ≤ c j 2 + d j 2 for 1 ≤ j ≤ n , and d G ( z k ) = 2 g k 2 ∈ [ g k , h k ] for 1 ≤ k ≤ l . One can see 2 a 2 ≥ a ≥ 2 a 2 for any nonnegative integer a . Thus, G is the graph that satisfies the conditions and so the proof is completed.□

Notice that the process of reconstructing a 3-partite graph is really complicated, since each case requires repeated verification to determine whether to apply. And as n increases, the construction process becomes more tedious. However, if the initial graph is an empty graph, the steps will become easier. In light of this, an example is given to demonstrate the construction process of a 3-partite graph.

Example

Take η 1 = ( [ 2 , 3 ] , [ 1 , 3 ] , [ 1 , 2 ] ) , η 2 = ( [ 2 , 4 ] , [ 1 , 2 ] ) , and η 3 = ( [ 0 , 6 ] ) , which satisfy a 1 ≥ a 2 ≥ a 3 , c 1 ≥ c 2 , and g 1 = 0 . It is easy to check that the conditions (1)–(6) all hold. Next, we proceed to construct a 3-partite graph that satisfies the requirements using the steps in the proof of Theorem 2.1.

We may start with the empty graph with partite sets X = { x 1 , x 2 , x 3 } , Y = { y 1 , y 2 } , and Z = { z 1 } . Clearly, r = 1 . Since y j ↮ x k for all j and all k > r , Case 1 does not need to be executed. Notice that d ( y 1 ) < d 1 2 = 2 and y 1 ↮ x 1 , apply Case 2 and connect y 1 and x 1 . So d ( x 1 ) = 1 = a 1 2 and d ( y 1 ) = 1 < d 1 2 = 2 . Increasing r by one and applying Case 2 yields y 1 ↔ x 2 . In this case, we obtain d ( x 2 ) = 1 = ⌈ a 2 2 ⌉ and d ( y 1 ) = d 1 2 = 2 . Now it comes to r = 3 , apply Case 2 and connect y 2 and x 3 . So we arrive at d ( x 3 ) = 1 = a 3 2 and d ( y 2 ) = d 2 2 = 1 . Thus, the graph G 1 ′ is obtained. In addition, it is clear that d ( y j ) = c j 2 for all j ≤ 2 , and there is no s such that d ( y s ) < c s 2 = 1 . Thus, Case 3 does not have to run anymore and so G 1 = G 1 ′ , whose degree sequence is ( d ( x 1 ) , d ( x 2 ) , d ( x 3 ) ; d ( y 1 ) , d ( y 2 ) ) = ( 1 , 1 , 1 ; 2 , 1 ) . Similarly, we can obtain the graphs G 2 , and G 3 , whose degree sequences are ( 1 , 1 , 1 ; 3 ) and ( 1 , 1 ; 2 ) , respectively. Finally, set G = G 1 ∪ G 2 ∪ G 3 , then G is the 3-partite graph that satisfies the requirements. A special case of G is given below. Let H be a 3-partite graph with V = { x 1 , x 2 , x 3 } ∪ { y 1 , y 2 } ∪ { z 1 } and E = { e 1 , e 2 , … , e 8 } , where e 1 = ( x 1 , z 1 ) , e 2 = ( x 2 , z 1 ) , e 3 = ( x 3 , z 1 ) , e 4 = ( x 1 , y 1 ) , e 5 = ( x 2 , y 1 ) , e 6 = ( x 3 , y 2 ) , e 7 = ( y 1 , z 1 ) , e 8 = ( y 2 , z 1 ) .

The conditions (1)–(6) of Theorem 2.1 are not necessary, as can be seen by taking η 1 = ( [ 2 , 3 ] , [ 0 , 2 ] ) , η 2 = ( [ 2 , 4 ] , [ 1 , 2 ] ) , and η 3 = ( [ 1 , 2 ] , [ 0 , 1 ] ) , which satisfy a 1 ≥ a 2 , c 1 ≥ c 2 and g 1 ≥ g 2 . Let F be a 3-partite graph with V = { x 1 , x 2 } ∪ { y 1 , y 2 } ∪ { z 1 , z 2 } , and E = { e 1 , e 2 , … , e 6 } , where e 1 = ( x 1 , y 1 ) , e 2 = ( x 1 , y 2 ) , e 3 = ( x 1 , z 1 ) , e 4 = ( x 2 , y 1 ) , e 5 = ( y 1 , z 1 ) , e 6 = ( y 1 , z 2 ) . It is easy to see that F is a realization of ( η 1 ; η 2 ; η 3 ) . But (5) is not true for q = 2 .

Theorem 2.2

Let η 1 = ( [ a 1 , b 1 ] , … , [ a m , b m ] ) , η 2 = ( [ c 1 , d 1 ] , … , [ c n , d n ] ) , and η 3 = ( [ g 1 , h 1 ] , … , [ g l , h l ] ) be three sequences of intervals with a 1 ≥ a 2 ≥ … ≥ a m , c 1 ≥ c 2 ≥ … ≥ c n and g 1 ≥ g 2 ≥ … ≥ g l . If ( η 1 ; η 2 ; η 3 ) is realized by a 3-partite graph, then

(8) ∑ i = 1 t a i ≤ ∑ j = 1 n min { d j , t } + ∑ j = 1 l min { h j , t } f o r e v e r y t w i t h 1 ≤ t ≤ m ,

(9) ∑ i = 1 p c i ≤ ∑ j = 1 m min { b j , p } + ∑ j = 1 l min { h j , p } f o r e v e r y p w i t h 1 ≤ p ≤ n ,

(10) ∑ i = 1 q g i ≤ ∑ j = 1 m min { b j , q } + ∑ j = 1 n min { d j , q } f o r e v e r y q w i t h 1 ≤ q ≤ l .

Proof

Let G be a realization of ( η 1 ; η 2 ; η 3 ) , and its partite sets are X , Y , and Z . Consider the edges incident to a set of t vertices in X . Each y j ∈ Y is incident to not more than t of these vertices, and also incident to not more than d Y X ( y j ) of these vertices, where d Y X ( y j ) is the maximum contribution to these t vertices in X from edges incident to y j ∈ Y . Analogous to the case of vertices in Z . Therefore,

∑ i = 1 t a i ≤ ∑ i = 1 t d G ( x i ) ≤ ∑ j = 1 n min { d Y X ( y j ) , t } + ∑ j = 1 l min { d Z X ( z j ) , t } ≤ ∑ j = 1 n min { d G ( y j ) , t } + ∑ j = 1 l min { d G ( z j ) , t } ≤ ∑ j = 1 n min { d j , t } + ∑ j = 1 l min { h j , t } .

Hence, (8) holds. Similarly, it can be proved that the conditions (9) and (10) are also valid.□

Theorems 2.1 and 2.2 can derive two corollaries, which are necessary condition and sufficient condition, and these two corollaries generalize Gale-Ryser theorem to 3-partite graph. However, the forgoing counterexample implies that one of the two conditions is sufficient but not necessary, and the other is necessary but not sufficient.

Let α = ( α 1 , α 2 , … , α m ) , β = ( β 1 , β 2 , … , β n ) and γ = ( γ 1 , γ 2 , … , γ l ) be three nonincreasing sequences. The triple ( α ; β ; γ ) is said to be realized by a 3-partite graph (i.e., tripartite graphic) if there exists a 3-partite graph G with tripartition X = { x 1 , x 2 , … , x m } , Y = { y 1 , y 2 , … , y n } and Z = { z 1 , z 2 , … , z l } so that d G ( x i ) = α i for i = 1 , 2 , … , m , d G ( y j ) = β j for j = 1 , 2 , … , n and d G ( z k ) = γ k for k = 1 , 2 , … , l .

Corollary 2.3

Let α = ( α 1 , α 2 , … , α m ) , β = ( β 1 , β 2 , … , β n ) , and γ = ( γ 1 , γ 2 , … , γ l ) be three nonincreasing sequences. If

∑ i = 1 s α i 2 ≤ ∑ j = 1 n min β j 2 , s f o r e v e r y s w i t h 1 ≤ s ≤ m

and

∑ i = 1 t α i 2 ≤ ∑ j = 1 l min γ j 2 , t f o r e v e r y t w i t h 1 ≤ t ≤ m

follow, then ( α ; β ; γ ) is realized by a 3-partite graph.

Corollary 2.4

Let α = ( α 1 , α 2 , … , α m ) , β = ( β 1 , β 2 , … , β n ) , and γ = ( γ 1 , γ 2 , … , γ l ) be three nonincreasing sequences. If ( α ; β ; γ ) is realized by a 3-partite graph, then for each s with 1 ≤ s ≤ m ,

∑ i = 1 s α i ≤ ∑ j = 1 n min { β j , s } + ∑ k = 1 l min { γ k , s } .

Theorem 2.5 provides a necessary condition for ( α ; β ; γ ) to be realized by a 3-partite graph, and the condition is stronger than the one in Corollary 2.4.

Theorem 2.5

∑ i = 1 s α i ≤ min ∑ j = 1 n β j − μ , n s + min ∑ k = 1 l γ k − μ , l s ,

where μ = 1 2 ∑ j = 1 n β j + ∑ k = 1 l γ k − ∑ i = 1 m α i .

Proof

Suppose that ( α ; β ; γ ) is realized by a 3-partite graph G with partite sets X = { x 1 , x 2 , … , x m } , Y = { y 1 , y 2 , … , y n } , and Z = { z 1 , z 2 , … , z l } , then d G ( x i ) = α i for i = 1 , 2 , … , m , d G ( y j ) = β j for j = 1 , 2 , … , n and d G ( z k ) = γ k for k = 1 , 2 , … , l . Denote μ by the number of edges of G between Y and Z . One can see that

μ = ∑ j = 1 n d ( y j ) − ∑ j = 1 n d X ( y j ) = ∑ j = 1 n β j − ∑ j = 1 n d X ( y j )

and

μ = ∑ k = 1 l d ( z k ) − ∑ k = 1 l d X ( z k ) = ∑ k = 1 l γ k − ∑ k = 1 l d X ( z k ) ,

where d X ( y j ) denotes the contribution of all vertices in X to vertex y j ∈ Y . Thus,

∑ j = 1 n d X ( y j ) = ∑ j = 1 n β j − ∑ k = 1 l γ k + ∑ k = 1 l d X ( z k )

and

∑ i = 1 m α i = ∑ i = 1 m d ( x i ) = ∑ j = 1 n d X ( y j ) + ∑ k = 1 l d X ( z k ) = ∑ j = 1 n β j − ∑ k = 1 l γ k + 2 ∑ k = 1 l d X ( z k ) .

After the transpose, we can obtain

∑ k = 1 l d X ( z k ) = 1 2 ∑ i = 1 m α i − ∑ j = 1 n β j + ∑ k = 1 l γ k

and then

μ = 1 2 ∑ j = 1 n β j + ∑ k = 1 l γ k − ∑ i = 1 m α i .

∑ j = 1 n d X ( y j ) = ∑ j = 1 n β j − μ , ∑ k = 1 l d X ( z k ) = ∑ k = 1 l γ k − μ .

Hence, for each s with 1 ≤ s ≤ m , we have

∑ i = 1 s α i = ∑ i = 1 s d ( x i ) ≤ ∑ j = 1 n min { d X ( y j ) , s } + ∑ k = 1 l min { d X ( z k ) , s } ≤ min ∑ j = 1 n d X ( y j ) , n s + min ∑ k = 1 l d X ( z k ) , l s = min ∑ j = 1 n β j − μ , n s + min ∑ k = 1 l γ k − μ , l s .

Thus, the conclusion is true.□

Based on the previous results, we want to establish a necessary and sufficient condition for ( α ; β ; γ ) to be realized by a 3-partite graph or a hypergraph. However, it seems difficult to develop such a characterization.

Problem 2.1

Determine a characterization of 3-partite graphic sequences.

Problem 2.2

Investigate necessary condition and sufficient condition for ( α ; β ; γ ) to be realized by an r -uniform hypergraph.

3 n -partite graphs

In addition to the results on 3-partite graphs in Section 2, we also investigate the case of n -partite graphs. Let η 1 = ( [ a 1 1 , b 1 1 ] , … , [ a k 1 1 , b k 1 1 ] ) , η 2 = ( [ a 1 2 , b 1 2 ] , … , [ a k 2 2 , b k 2 2 ] ) , … , η n = ( [ a 1 n , b 1 n ] , … , [ a k n n , b k n n ] ) be n sequences of intervals with a 1 1 ≥ a 2 1 ≥ … ≥ a k 1 1 , …, a 1 n ≥ a 2 n ≥ … ≥ a k n n . The n -tuple ( η 1 ; η 2 ; … ; η n ) is said to be realized by an n -partite graph if there exists an n -partite graph G with partition X 1 = { x 1 1 , x 2 1 , … , x k 1 1 } , X 2 = { x 1 2 , x 2 2 , … , x k 2 2 } , … , X n = { x 1 n , x 2 n , … , x k n n } so that a i 1 ≤ d G ( x i 1 ) ≤ b i 1 for i = 1 , 2 , … , k 1 , … , a i n ≤ d G ( x i n ) ≤ b i n for i = 1 , 2 , … , k n .

Theorem 3.1

Let η 1 = ( [ a 1 1 , b 1 1 ] , … , [ a k 1 1 , b k 1 1 ] ) , η 2 = ( [ a 1 2 , b 1 2 ] , … , [ a k 2 2 , b k 2 2 ] ) , … , η n = ( [ a 1 n , b 1 n ] , … , [ a k n n , b k n n ] ) be n sequences of intervals with a 1 1 ≥ a 2 1 ≥ … ≥ a k 1 1 , … , a 1 n ≥ a 2 n ≥ … ≥ a k n n . If the following n ( n − 1 ) 2 inequalities hold:

∑ i = 1 r 1 a i 1 n − 1 ≤ min ∑ j = 1 k s min b j s n − 1 , r 1 , s = 2 , 3 , … , n , 1 ≤ r 1 ≤ k 1 , ⋯ ⋯ ∑ i = 1 r m a i m n − 1 ≤ min ∑ j = 1 k s min b j s n − 1 , r m , s ∈ { 1 , … , n } \ m , 1 ≤ r m ≤ k m , ⋯ ⋯ ∑ i = 1 r n a i n n − 1 ≤ min ∑ j = 1 k s min b j s n − 1 , r n , s = 1 , 2 , … , n − 1 , 1 ≤ r n ≤ k n ,

then ( η 1 ; η 2 ; … ; η n ) is realized by an n-partite graph.

Proof

To prove that ( η 1 ; η 2 ; … ; η n ) is realized by an n -partite graph, it suffices to construct an n -partite graph G that satisfies the requirements. We first construct n 2 bipartite graphs G 12 , G 13 , … , G 1 n , G 23 , G 24 , … , G 2 n , … , G ( n − 1 ) n , and then set G = G 12 ∪ G 13 ∪ … ∪ G ( n − 1 ) n . It is not hard to obtain these bipartite graphs by using the constructive approach in the proof of Theorem 2.1.

G 12 : a i 1 n − 1 ≤ d X 2 ( x i 1 ) ≤ b i 1 n − 1 ( 1 ≤ i ≤ k 1 ) , d X 1 ( x j 2 ) = a j 2 n − 1 ( 1 ≤ j ≤ k 2 ) ; G 13 : a i 1 n − 1 ≤ d X 3 ( x i 1 ) ≤ b i 1 n − 1 ( 1 ≤ i ≤ k 1 ) , d X 1 ( x j 3 ) = a j 3 n − 1 ( 1 ≤ j ≤ k 3 ) ; ⋯ ⋯ G 1 n : a i 1 n − 1 ≤ d X n ( x i 1 ) ≤ b i 1 n − 1 ( 1 ≤ i ≤ k 1 ) , d X 1 ( x j n ) = a j n n − 1 ( 1 ≤ j ≤ k n ) ; G 23 : a i 2 n − 1 ≤ d X 3 ( x i 2 ) ≤ b i 2 n − 1 ( 1 ≤ i ≤ k 2 ) , d X 2 ( x j 3 ) = a j 3 n − 1 ( 1 ≤ j ≤ k 3 ) ; ⋯ ⋯ G 2 n : a i 2 n − 1 ≤ d X n ( x i 2 ) ≤ b i 2 n − 1 ( 1 ≤ i ≤ k 2 ) , d X 2 ( x j n ) = a j n n − 1 ( 1 ≤ j ≤ k n ) ; ⋯ ⋯ G ( n − 1 ) n : a i n − 1 n − 1 ≤ d X n ( x i n − 1 ) ≤ b i n − 1 n − 1 ( 1 ≤ i ≤ k n − 1 ) , d X n − 1 ( x j n ) = a j n n − 1 ( 1 ≤ j ≤ k n ) .

Thus, for each i ∈ ⟨ k 1 ⟩ = { 1 , 2 , … , k 1 } , we have

d G ( x i 1 ) = d X 2 ( x i 1 ) + d X 3 ( x i 1 ) + … + d X n ( x i 1 ) ∈ ( n − 1 ) a i 1 n − 1 , ( n − 1 ) b i 1 n − 1 ⊆ [ a i 1 , b i 1 ] ;

for each i ∈ ⟨ k 2 ⟩ , we have

d G ( x i 2 ) = d X 1 ( x i 2 ) + d X 3 ( x i 2 ) + … + d X n ( x i 2 ) ∈ ( n − 1 ) a i 2 n − 1 , a i 2 n − 1 + ( n − 2 ) b i 2 n − 1 ⊆ [ a i 2 , b i 2 ] ; ⋯ ⋯

for each i ∈ ⟨ k n ⟩ , we have

d G ( x i n ) = d X 1 ( x i n ) + d X 2 ( x i n ) + … + d X n − 1 ( x i n ) = ( n − 1 ) a i n n − 1 ⊆ [ a i n , b i n ] .

Therefore, the graph G is the n -partite graph that satisfies the requirements. Hence, ( η 1 ; η 2 ; … ; η n ) is realized by an n -partite graph.□

Theorem 3.2

Let η 1 = ( [ a 1 1 , b 1 1 ] , … , [ a k 1 1 , b k 1 1 ] ) , η 2 = ( [ a 1 2 , b 1 2 ] , … , [ a k 2 2 , b k 2 2 ] ) , … , η n = ( [ a 1 n , b 1 n ] , … , [ a k n n , b k n n ] ) be n sequences of intervals with a 1 1 ≥ a 2 1 ≥ … ≥ a k 1 1 , … , a 1 n ≥ a 2 n ≥ … ≥ a k n n . If ( η 1 ; η 2 ; … ; η n ) is realized by an n-partite graph, then

∑ i = 1 w a i 1 ≤ ∑ j = 1 k 2 min { b j 2 , w } + … + ∑ j = 1 k n min { b j n , w } , 1 ≤ w ≤ k 1 , ⋯ ⋯ ∑ i = 1 w a i n ≤ ∑ j = 1 k 1 min { b j 1 , w } + … + ∑ j = 1 k n − 1 min { b j n − 1 , w } , 1 ≤ w ≤ k n .

Proof

According to symmetry, we just have to prove the first inequality by induction on n . When n = 2 , it is clearly true. Now suppose that the inequality is true for any n that is less than or equal to t . Then for each w with 1 ≤ w ≤ k 1 , we derive

∑ i = 1 w a i 1 ≤ ∑ j = 1 k 2 min { b j 2 , w } + … + ∑ j = 1 k t min { b j t , w } < ∑ j = 1 k 2 min { b j 2 , w } + … + ∑ j = 1 k t min { b j t , w } + ∑ j = 1 k t + 1 min { b j t + 1 , w } .

Thus, for n = t + 1 , the inequality is also true, and hence, the proof is complete.□

It follows from Theorems 3.1 and 3.2 that Corollaries 3.3 and 3.4 are true.

Corollary 3.3

Let η 1 = ( α 1 1 , α 2 1 , … , α k 1 1 ) , η 2 = ( α 1 2 , α 2 2 , … , α k 2 2 ) , … , η n = ( α 1 n , α 2 n , … , α k n n ) be n nonincreasing sequences. If for each w with 1 ≤ w ≤ k 1 ,

∑ i = 1 w α i 1 n − 1 ≤ min ∑ j = 1 k s min α j s n − 1 , w , s = 2 , 3 , … , n ,

then ( η 1 ; η 2 ; … ; η n ) is realized by an n-partite graph.

Corollary 3.4

Let η 1 = ( α 1 1 , α 2 1 , … , α k 1 1 ) , η 2 = ( α 1 2 , α 2 2 , … , α k 2 2 ) , … , η n = ( α 1 n , α 2 n , … , α k n n ) be n nonincreasing sequences. If ( η 1 ; η 2 ; … ; η n ) is realized by an n-partite graph, then for each w with 1 ≤ w ≤ k 1 ,

∑ i = 1 w α i 1 ≤ ∑ j = 1 k 2 min { α j 2 , w } + … + ∑ j = 1 k n min { α j n , w } .

4 Conclusion

To sum up, we obtain the sufficient condition and necessary condition for three sequences of intervals to be realized by a 3-partite graph. As corollaries, we obtain the sufficient condition and necessary condition for the triple ( α ; β ; γ ) to be realized by a 3-partite graph, which extend Gale-Ryser theorem. Besides, we also present the sufficient condition and necessary condition for n sequences of intervals to be realized by an n -partite graph. As for the possible future research goals, we would like to study the polynomial time algorithm, learn some programming languages, and try to design a new algorithm to reconstruct bipartite graphs and 3-partite graphs.

Acknowledgements

The authors are grateful for the reviewers’ valuable suggestions and comments.

Funding information: This work was supported by the National Natural Science Foundation of China (NSFC11961019) and Hainan Provincial Natural Science Foundation of China (Grant Nos. 122RC545 and 621RC510).
Author contributions: All authors have accepted responsibility for the entire content of the manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. JG prepared the material, analyzed the data and proved the results. HL typed the manuscript. MF, YZ, and LL polished the manuscript.
Conflict of interest: The authors state no conflict of interest.

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Received: 2024-02-04

Revised: 2024-09-28

Accepted: 2024-10-10

Published Online: 2024-12-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0093

Keywords for this article

tripartite graphs; monotonous degree condition; n-partite graphs

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