Path homology theory of edge-colored graphs

1 Introduction

In this paper, we continue to study the homological properties of colored digraphs and graphs. It should be mentioned that, on the basic concepts of the path homology theory introduced in [1,2,3], the collection of path homology theories for vertex colored (di)graphs have been already constructed in [4].

The path homology theory is a homology theory for digraphs that computes the simplicial homology of a finite simplicial complex S if applied to its incidence digraph G = G S defined in the following way (see [1,3]). For a finite simplicial complex S , let V be the set of its simplexes. Consider a digraph G = ( V , E ) with the set of vertices V as above and the set of arrows E such that σ → τ is an arrow if and only if τ ⊂ σ . Then the path homology of the digraph G is naturally isomorphic to the simplicial homology of the simplicial complex S . The cohomology theory of digraphs that is dual to the path homology theory was introduced in previous studies [5,6, 7,8]. This cohomology theory was motivated by the physical applications of discrete mathematics. This theory provides a differential calculus on digraphs and discrete sets which are considered as discretizations of topological spaces. Thus, various physical theories can be formulated on a discrete set analogous to the continuum case.

Recently, the path homology theory has been used in applications of the persistent homology to the various types of networks (see, for e.g., [9,10]). So in [10] the directed networks related to applications are considered and efficient algorithms for computing one-dimensional path homology and its persistent version are developed.

In what follows, we provide the category of edge-colored digraphs together with the notion of the homotopy and the homotopy category of edge-colored digraphs. For any edge-colored digraph, we define the collection of edge-colored path homology groups and indicate the possibility of the functorial passage from the category of graphs to the subcategory of symmetrical digraphs. We also prove the colored homotopy invariance of the colored path homology and describe its algebraic properties. More precisely, we discuss the natural filtration of the path homology groups of the edge-colored digraph, construct commutative diagrams and braids of exact sequences for those homology groups, and describe the spectral sequence that is associated with the filtration. This paper contains many examples to illustrate the non-triviality of edge-colored homology groups. We discuss also possible applications of the constructed theory.

2 Preliminaries

In this section, for the sake of convenience, we review some basic notions of various categories of digraphs, graphs (see [11,12,13]), and the path homology theory (see [1,2,3]).

A graph G = ( V , E ) is a nonempty set V of objects called vertices together with a set E of non ordered pairs { v , w } ∈ E of distinct vertices v , w ∈ V called edges.

A directed graph or simply a digraph G = ( V , E ) is a nonempty set V of objects called vertices together with a set E of ordered pairs of distinct vertices of V called directed edges or arrows. A pair ( v , w ) ∈ E is denoted v → w , whereas the vertices v and w are called, respectively, the origin and the end of the given arrow. Accordingly, we write v = orig ( v → w ) , w = end ( v → w ) .

Two different edges (arrows) e , e ′ ∈ E G of a graph (digraph) G are called incident if they have at least one common vertex.

Let G = ( V , E ) be a graph. The corresponding symmetric digraph G ˜ = ( V ˜ , E ˜ ) is built from the same set of vertices V ˜ = V but each edge { v , w } ∈ E gives rise to two arrows, namely ( v → w ) , ( w → v ) ∈ E ˜ .

Let G = ( V G , E G ) and H = ( V H , E H ) be any graphs. A mapping f : G → H between two graphs is defined as a mapping f : V G → V H such that for any edge { v , w } ∈ E G , we have either { f ( v ) , f ( w ) } ∈ E H or f ( v ) = f ( w ) ∈ V H . The mapping f is called a homomorphism if { f ( v ) , f ( w ) } ∈ E H for any { v , w } ∈ E G .

Let G = ( V G , E G ) and H = ( V H , E H ) be any digraphs this time. A digraph mapping f : G → H (or simply a mapping) from a digraph G to a digraph H is a mapping f : V G → V H such that for any arrow ( v → w ) ∈ E G , we have either ( f ( v ) → f ( w ) ) ∈ E H or f ( v ) = f ( w ) ∈ V H . The mapping f is called a homomorphism if ( f ( v ) → f ( w ) ) ∈ E H for any ( v → w ) ∈ E G .

Having in mind all the considerations above, we point out that we are thus provided with the following categories: the category G of graphs and graph mappings, the subcategory NG of G with the same objects and with the morphisms given by homomorphisms, the category D of digraphs and digraph mappings, and the subcategory ND of D with the same objects and with the morphisms given by homomorphisms. It is easy to check that the passing from a graph G to a symmetric digraph G ˜ naturally defines the mapping of morphisms and gives the functors S : G → D and NS : NG → ND .

Let G and H be any graphs. We define their Box product G □ H as a graph with the set of vertices V G □ H = V G × V H and the set of edges E G □ H such that { ( x , y ) , ( x ′ , y ′ ) } ∈ E G □ H if and only if either x = x ′ and { y , y ′ } ∈ E H or { x , x ′ } ∈ E G and y = y ′ .

We define the Box product G □ H of two digraphs G and H in a similar fashion, namely as a digraph with the set of vertices V G □ H = V G × V H and the set of edges E G □ H such that there is an arrow ( ( x , y ) → ( x ′ , y ′ ) ) ∈ E G □ H if and only if either x = x ′ and y → y ′ or x → x ′ and y = y ′ .

Fix n ≥ 0 . Denote by J n a graph with the set of vertices { 0 , 1 , … , n } and the set of edges { { i , i + 1 } ∣ 0 ≤ i ≤ n − 1 } . From now on J n will be called a segment graph. Denote by I n any digraph with the set of vertices { 0 , 1 , … , n } and the set of edges in which there is exactly one of the arrows: i → i + 1 or i + 1 → i for 0 ≤ i ≤ n − 1 . Such I n will be called a segment digraph.

Two graph mappings f 0 , f 1 : G → H are called homotopic and denoted accordingly f 0 ≃ f 1 if there exists a segment graph J n and a mapping F : G □ J n → H called a homotopy between f 0 and f 1 such that

Two graph homomorphisms are called strongly homotopic if the homotopy between them is also a homomorphism.

Two digraph mappings f 0 , f 1 : G → H are called homotopic and denoted f 0 ≃ f 1 if there exists a segment digraph I n together with a mapping F : G □ I n → H called a homotopy between f 0 and f 1 such that

Two digraph homomorphisms are called strongly homotopic if the homotopy between them is a homomorphism. In the case of homotopy F for which n = 1 , the map F is called the one-step homotopy of (di)graphs.

Now we give a brief explanation of the notion of homotopy in the case of graphs that is similar to the case of digraph given in [2]. There exists a one-step homotopy F between graph mappings f : G → H and g : G → H if and only if either f ( x ) = g ( x ) ∈ V G or { f ( x ) , g ( x ) } ∈ E H for every x ∈ V G . It follows that two graph mappings f and g are homotopic if there is a finite sequence of graph mappings f = f 0 , f 1 , … , f n = g from G to H such that f k and f k + 1 are one-step homotopic (see also Examples 3.6, 3.7, and 3.9).

This time, it is easy to check that we are provided with the following categories: the category G ′ of graphs and classes of homotopic mappings, the category NG ′ with the same objects and with the morphisms given by the classes of strongly homotopic homomorphisms, the category D ′ of digraphs and classes of homotopic mappings, and the category ND ′ with the same objects and with the morphisms given by the classes of strongly homotopic homomorphisms.

It follows easily from [2, Proposition 6.5] that the functors S and NS preserve the relation to be homotopic and hence define functors S ′ : G ′ → D ′ and NS ′ : NG ′ → ND ′ .

Now we turn our attention to the path homology groups of any (di)graph and provide necessary definitions. Let G = ( V , E ) be a digraph and R be a commutative ring.

An elementary p -path denoted e i 0 … i p is defined as any sequence i 0 , … , i p of vertices. Let Λ p ( V , R ) be a free R -module generated by all elementary p -paths. The elements of Λ p are called p -paths. We set Λ − 1 = 0 and define the boundary operator ∂ : Λ p → Λ p − 1 in the following way:

The operator defined above has the property ∂ 2 = 0 .

For p ≥ 1 , let I p = I p ( V ) be the submodule of Λ p that is generated by all elementary paths for which two consecutive vertices are equal. Set I 0 = I − 1 = 0 . Then ∂ ( I p ) ⊂ I p − 1 and we obtain a chain factor complex ℛ ∗ = ℛ p ( V , R ) = Λ p / I p with the differential that is induced by ∂ . The elements of this module are called regular paths and regular elementary paths for basic elements. Now we return to the consideration of the digraph G = ( V , E ) . For p ≥ 1 , a regular elementary path e i 0 … i p is called allowed if ( i k → i k + 1 ) is an arrow of the digraph G for 0 ≤ k ≤ p − 1 . For p ≥ 0 , let A p = A p ( G , R ) be a submodule of ℛ p ( V , R ) that is generated by all the allowed elementary p -paths and set A − 1 = 0 . Define a submodule Ω p = Ω p ( G , R ) = { v ∈ A p : ∂ v ∈ A p − 1 } of the module A p . The submodule Ω p consists of all linear combinations v of allowed paths for which ∂ v is a linear combination of allowed paths as well. Having all the above in mind, we obtain a chain complex Ω ∗ ( G , R ) . The homologies of this chain complex are called path homologies of the digraph G and denoted by

For a graph G we define the path homology H p ( G , R ) setting H p ( G , R ) ≔ H p ( G ˜ , R ) , where G ˜ denotes the corresponding symmetric digraph.

For a digraph mapping f : G → H , define for every p ≥ 0 the induced map f ∗ : Λ p ( V G ) → Λ p ( V H ) given on the basic elements by the rule

The map f ∗ is a morphism of chain complexes and it is clear that f ∗ ( I p ( V G ) ) ⊂ I p ( V H ) [2]. Hence, we obtain an induced chain map of quotient chain complexes f ∗ : ℛ ∗ ( V G ) → ℛ ∗ ( V H ) that is defined on basic elements by the rule

It follows from (2.2) that if a path e i 0 … i p ∈ A p ( G ) , then f ∗ ( e i 0 … i p ) is 0 ∈ A p ( H ) or is allowed. Hence, f ∗ ( A p ( G ) ) ⊂ A p ( H ) . Now the standard line of arguments provides a morphism of chain complexes Ω ∗ ( G ) → Ω ∗ ( H ) and a homomorphism of homology groups H ∗ ( G , R ) → H ∗ ( H , R ) .

Thus, the path homology groups of digraphs and graphs are functorial and these groups are homotopy invariant [2].

3 Categories of edge-colored digraphs and graphs

In this section, we introduce several categories of edge-colored graphs and digraphs, describe their basic properties, and provide examples. In the next section, we shall use these categories for constructing the collection of edge-colored path homology theories.

An edge coloring of a graph (digraph) G = ( V G , E G ) is given by an assignment of a color to each edge (arrow) e ∈ E G . An edge coloring is called proper if incident edges (arrows) have distinct colors. An edge coloring is called k-improper if for any edge (arrow) e ∈ E G there exist at most k incident edges (arrows) having the same color as e . An edge coloring that uses k colors is called a k-edge coloring.

An edge coloring of a (graph) digraph G = ( V G , E G ) can be considered as a pair ( G , φ ) , where φ : E G → N is a function. For the k -edge coloring, we assume that φ : E G → { 1 , … , k } . In what follows, we shall consider proper coloring as the k -improper coloring with k = 0 . Since from now on only edge colorings will be considered, the word edge will be accordingly omitted.

Thus, we obtain the following categories: the category CG of colored graphs and the colored morphisms, the subcategory CNG with the same objects and with the colored morphisms that are homomorphisms, the category CD of colored digraphs and the colored morphisms, and the subcategory CND with the same objects and with the colored morphisms given by homomorphisms.

For a colored graph ( G , φ ) , we define a colored symmetric digraph ( G ˜ , φ ˜ ) by setting φ ˜ ( v → w ) = φ ˜ ( w → v ) = φ ( v , w ) . Thus, as in Section 2, we obtain the functors CS : CG → CD and CNS : CNG → CND .

For a colored (di)graph ( G , φ ) , we can consider this (di)graph G without any coloring. Any morphism of colored (di)graphs f : ( G , φ ) → ( H , ψ ) is, in particular, a (di)graph mapping and we obtain a collection of forgetful functors from the categories of colored (di)graphs to the corresponding categories of (di)graphs.

For a colored digraph ( G , φ ) , define a function κ which returns the number of different colors that are being used for the coloring of arrows in a non empty set A of arrows in E G . For every allowed path v = e i 0 … i p in the colored digraph ( G , φ ) , we define κ ( v ) = 0 for p = 0 and, for p ≥ 1 , we put κ ( v ) = κ ( A ) , where A consists of arrows i 0 → i 1 , … , i p − 1 → i p of the path v . An allowed regular elementary path v = e i 0 … i p ( p ≥ 0 ) is called s-colored if κ ( v ) = s .

Let f : ( G , φ ) → ( H , ψ ) be a colored morphism. By (2.2) and Definition 3.1, for a regular path v = e i 0 … i p in the digraph G we have κ ( f ∗ ( v ) ) = κ ( v ) if the path f ∗ ( v ) is regular. In the opposite case of the non-regular path f ∗ ( v ) , we set κ ( f ∗ ( v ) ) = 0 . Then we have κ ( f ∗ ( v ) ) ≤ κ ( v ) for any allowed path v = e i 0 … i p in the colored digraph ( G , φ ) .

Now we introduce the notion of a s-colored homotopy between two colored morphisms of digraphs. Denote by I the segment digraph I 1 = ( 0 → 1 ) . For any allowed elementary p -path v = e i 0 … i p in a colored digraph ( G , φ ) , define an allowed elementary ( p + 1 ) -path v ^ k ( 0 ≤ k ≤ p ) in G □ I by v ^ k = e i 0 … i k i k ′ … i p ′ , where i j denotes the vertex ( i j , 0 ) ∈ G □ { 0 } and i j ′ denotes the vertex ( i j , 1 ) ∈ G □ { 1 } . These notions are well defined since we have the natural identifications G = G □ { 0 } and G = G □ { 1 } . For the product G □ I , we define a coloring φ i of the subgraph G □ { i } ⊂ G □ I ( i = 0 , 1 ) by setting φ i [ ( v , i ) → ( w , i ) ] = φ ( v → w ) , where ( ( v , i ) → ( w , i ) ) ∈ E G □ { i } . Denote by E [ 0 , 1 ] the set of edges of the digraph G □ I that have the form ( ( v , 0 ) → ( v , 1 ) ) with v ∈ V G . Any coloring φ [ 0 , 1 ] : E [ 0 , 1 ] → N of the set of edges E [ 0 , 1 ] together with the colorings φ i for i = 0 , 1 induce the coloring of the digraph G □ I which we denote Φ [ 0 , 1 ] : E G □ I → N .

Let f , g be s -colored homotopic morphisms as in Definition 3.4 with the sequence f , f 1 , … , f p = g of s -colored one-step homotopies. By Definition 3.3, for every pair ( f k , f k + 1 ) of consequent morphisms, we have two possibilities. First, there is a colored morphism F k : ( G □ I , Φ [ 0 , 1 ] k ) → ( H , ψ ) such that F k ∣ G □ { 0 } = f k and F k ∣ G □ { 1 } = f k + 1 . Second, there is a colored morphism F k : ( G □ I , Φ [ 0 , 1 ] k ) → ( H , ψ ) such that F k ∣ G □ { 0 } = f k + 1 and F k ∣ G □ { 0 } = f k . Now, define a segment digraph I p in the following way. For every pair of vertices ( k , k + 1 ) with 0 ≤ k ≤ p − 1 , there is an arrow k → k + 1 if the first case of the colored morphism F k occurs and there is an arrow k + 1 → k if the second case of the colored morphism F k occurs. Now, the coloring Φ [ 0 , 1 ] k is well defined on the subgraph k → k + 1 or the subgraph k + 1 → k of the digraph I p whichever case occurs. The colorings Φ [ 0 , 1 ] k with 0 ≤ k ≤ p − 1 define a coloring Φ of the digraph G □ I p . Moreover, the union of morphisms F k defines a colored morphism F : ( G □ I p , Φ ) → ( H , ψ ) since F k ∣ G □ { k + 1 } = F k + 1 ∣ G □ { k } . Please note also that the condition (i) of Definition 3.3 is satisfied for the morphism F k on the sub-digraph k → k + 1 (or the sub-digraph k + 1 → k in the second case) of I p under natural identification of k → k + 1 with I ( k + 1 → k with I , respectively).

We would like to indicate here that the notion of s -colored homotopy essentially depends on the number s = 0 , 1 , 2 … . Two colored digraph morphisms f and g are 0-colored homotopic only in the case of f = g . Indeed, let ( F , φ [ 0 , 1 ] ) be a 0-colored homotopy from f to g and f ( i ) ≠ g ( i ) for a vertex i ∈ V G . Then, for v = e i , we have κ ( v ) = 0 and κ ( F ∗ ( v ^ 0 ) ) = 1 . Thus, we obtain a contradiction and hence f ( i ) = g ( i ) for every i ∈ V G . Now let f , g : G → H be k -colored digraphs and s ≥ k . Then f and g are s -colored homotopic if and only if these morphisms are homotopic as digraph mappings. It follows from Definitions 3.3, 3.4, and the definition of k -coloring that for any regular paths v in G , we have κ ( v ) ≤ s and κ ( F ∗ ( v ^ k ) ) ≤ s .

Now we prove that relation to be s -colored homotopic is an equivalence relation and provide several examples.

The definition of s -colored homotopy in the category of colored graphs is similar and it is an equivalence relation on the set of colored morphisms of colored graphs.

From the considerations above it follows that, for s ≥ 0 , we are provided with the collection of s-colored homotopy categories of graphs C s G ′ in which the objects are colored graphs and morphisms are classes of s -colored homotopic morphisms. Similarly, we also obtain the collection of s -colored homotopy categories of digraphs C s D ′ and the s -colored homotopy categories C s NG ′ , C s ND ′ .

Below, we give an example of a non trivial one-step two-colored homotopy between colored morphisms of three-colored digraphs.

Figure 1

The edge-colored digraphs G and H , respectively.

Now we turn our attention to C s ND and provide an example of non trivial homotopy in this category for s = 2 .

Figure 2

The edge-colored digraph G and H , respectively.

4 Path homology of colored graphs and digraphs

In this section, we construct a collection of path homology theories defined on various categories of colored graphs and digraphs and describe their basic properties. To illustrate the definitions thus introduced, we also provide several examples. Let ( G , φ ) be a colored digraph. Recall that for every nonempty set of arrows A ⊂ E G we have defined the function κ which returns the number of different colors that are being used in the coloring of arrows from the set A .

Note that the set of all the elementary 0-colored paths coincides with the set of vertices V G . For k , p ≥ 1 , we define a free R -module ℒ p k = ℒ p k ( G , R ) = ℒ p k ( G , φ ) as a submodule of A p ( G , R ) generated by all the allowed regular elementary s -colored paths with 1 ≤ s ≤ k . Let ℒ p 0 = ℒ p 0 ( G ) = 0 for p ≥ 1 and ℒ − 1 k = 0 ,

for k ≥ 0 . Note that ℒ 0 k is the free R -module generated by all the elementary 0-colored paths. Thus, we obtain submodules ℒ p k ⊂ A p ( G , R ) ⊂ ℛ p ( V G , R ) for p ≥ − 1 and k ≥ 0 .

For p ≥ 1 , let