# On Path Homology of Vertex Colored (Di)Graphs

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- If $f\in Hom(A,B)$, $g\in Hom(B,C)$ and $h\in Hom(C,D)$, then $h\left(gf\right)=\left(hg\right)f\in Hom(A,D)$.
- (ii)
- For every object B there is a morphism ${Id}_{B}:B\to B$, such that if $f\in Hom(A,B)$, then ${Id}_{B}f=f$ and if $h\in Hom(B,C)$, then $h{Id}_{B}=h$.

- (i)
- $\mathcal{F}\left({Id}_{B}\right)={Id}_{\mathcal{F}\left(B\right)}$,
- (ii)
- $\mathcal{F}\left(gf\right)=\mathcal{F}\left(g\right)\mathcal{F}\left(f\right)$ for $f\in Hom(A,B)$, $g\in Hom(B,C)$.

- (i)
- ${f}_{i}\left(v\right)={f}_{i+1}\left(v\right)$ or ${f}_{i}\left(v\right)\to {f}_{i+1}\left(v\right)$ in the digraph H for arbitrary vertex v of the digraph G,
- (ii)
- ${f}_{i}\left(v\right)={f}_{i+1}\left(v\right)$ or ${f}_{i+1}\left(v\right)\to {f}_{i}\left(v\right)$ in the digraph H for arbitrary vertex v of the digraph G.

## 2. The Path Homology Theory for Digraphs

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Example**

**1.**

- (i)
- For the digraph $G=({V}_{G},{E}_{G})$ presented on the diagram below$$\begin{array}{ccccc}a& \u27f6& b& \u27f5& c\\ \downarrow & \phantom{\rule{1.em}{0ex}}& \uparrow \downarrow & \nearrow \\ d& \u27f5& e& \phantom{\rule{1.em}{0ex}}\end{array}$$we have ${V}_{G}=\{a,b,c,d,e\}$, whereas ${E}_{G}=\{a\to b,a\to d,e\to d,e\to b,b\to e,e\to c,c\to b\}$.
- (ii)
- Let G be a digraph given in (1) and $H=({V}_{H},{E}_{H})$ be a digraph provided below$$\begin{array}{ccccc}\phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& 3& \u27f5& 4\\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& \uparrow \downarrow & \nearrow \\ 1& \u27f5& 2& \phantom{\rule{1.em}{0ex}}\end{array}$$Let $f:{V}_{G}\to {V}_{H}$ be the mapping given by $f\left(a\right)=2,f\left(d\right)=1,f\left(b\right)=3,f\left(e\right)=2,f\left(c\right)=3$. Then the mapping f is the digraph mapping but it is not a non-degenerate digraph mapping since $f(c\to b)$ is not an arrow. Let $g:{V}_{G}\to {V}_{H}$ be the mapping given by $f\left(a\right)=2,f\left(d\right)=1,f\left(b\right)=3,f\left(e\right)=2,f\left(c\right)=4$. Then the mapping g is the non-degenerate digraph mapping.
- (iii)
- The explanation of the Box product is given in the diagram below:$$\begin{array}{ccccc}{y}^{\prime}\bullet & \phantom{\rule{1.em}{0ex}}& \stackrel{\left(x,{y}^{\prime}\right)}{\bullet}\phantom{\rule{1.em}{0ex}}& \u27f6& \stackrel{\left({x}^{\prime},{y}^{\prime}\right)}{\bullet}\\ \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\uparrow \phantom{\rule{4pt}{0ex}}& \phantom{\rule{1.em}{0ex}}& \uparrow & \phantom{\rule{1.em}{0ex}}& \uparrow \\ y\bullet & \phantom{\rule{1.em}{0ex}}& \stackrel{\left(x,y\right)}{\bullet}& \u27f6\phantom{\rule{1.em}{0ex}}& \stackrel{\left({x}^{\prime},y\right)}{\bullet}\\ \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}& \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\\ H\u2215G& \phantom{\rule{1.em}{0ex}}& \underset{x}{\bullet}& \u27f6& \underset{{x}^{\prime}}{\bullet}\phantom{\rule{4pt}{0ex}}\end{array}$$
- (iv)
- Let $G=({V}_{G},{V}_{H})$ be the digraph presented on the diagram$$\begin{array}{ccc}c& \u27f6& d\\ \uparrow & \phantom{\rule{1.em}{0ex}}& \uparrow \\ a& \u27f6& b\end{array}$$$$\begin{array}{c}F(a,0)=F(c,0)=0,F(b,0)=F(d,0)=1,\\ F(a,1)=F(c,1)=F(b,1)=F(d,1)=1,\\ F(a,2)=F(b,2)=0,F(c,2)=F(d,2)=1,\end{array}$$

**Theorem**

**1**

**.**The homology groups defined above are homotopy invariant and functorial for digraph mappings.

## 3. Categories of Colored Digraphs

**Definition**

**5.**

**Definition**

**6.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Example**

**2.**

- (i)
- Let G be the 1-improper colored digraph presented on (10).$$\begin{array}{ccccc}\begin{array}{c}\hfill {a}\end{array}& \phantom{\rule{1.em}{0ex}}& \to & \phantom{\rule{1.em}{0ex}}& \begin{array}{c}\hfill {b}\end{array}\\ \phantom{\rule{1.em}{0ex}}& \searrow & \phantom{\rule{1.em}{0ex}}& \nearrow \\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& \begin{array}{c}\hfill {c}\end{array}& \phantom{\rule{1.em}{0ex}}\end{array}$$$$f(a,0)=f(a,1)=a,\phantom{\rule{4pt}{0ex}}f(c,0)=c,\phantom{\rule{4pt}{0ex}}f(b,0)=b,\phantom{\rule{4pt}{0ex}}f(c,1)=f(b,1)=b.$$
- (ii)
- Let G be the proper colored digraph presented on (11):$$\begin{array}{ccc}{a}& \to & {d}\\ \uparrow & \phantom{\rule{1.em}{0ex}}& \uparrow \\ {a}& \to & {c}\end{array}$$$${f}_{1}\left(a\right)=a,\phantom{\rule{4pt}{0ex}}{f}_{1}\left(b\right)={f}_{1}\left(c\right)=c,\phantom{\rule{4pt}{0ex}}{f}_{1}\left(d\right)=d.$$
- (iii)
- Using the same line of arguments as in the case (ii) and the results from [2] (Section 3) it is possible to construct a lot of similar examples.

## 4. Path Homology of Colored Digraphs

**Definition**

**7.**

**Example**

**3.**

**Proposition**

**3.**

**Proof.**

**Example**

**4.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

## 5. Spectral Sequence for Path Homology Groups of a Colored Digraph

**Theorem**

**4.**

- 1.
- ${\mathcal{K}}_{*}^{p}=0$ for $p<0$.
- 2.
- There is a short exact sequence of chain complexes$$0\to {\mathcal{K}}_{*}^{p-1}\to {\mathcal{K}}_{*}^{p}\to {\mathcal{K}}_{*}^{p}/{\mathcal{K}}_{*}^{p-1}\to 0$$
- 3.
- $$\bigcup _{p\ge 0}{\mathcal{K}}_{*}^{p}={\mathcal{K}}_{*}.$$

**Proof.**

**Corollary**

**3.**

**Proposition**

**4.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Example**

**5.**

**Proposition**

**5.**

## 6. Path Homology of Colored Graphs

**Definition**

**8.**

- (i)
- A graph $\mathbf{G}$ is given by a set ${\mathbf{V}}_{\mathbf{G}}$ of vertices and a subset ${\mathbf{E}}_{\mathbf{G}}\subset {\mathbf{V}}_{\mathbf{G}}\times {\mathbf{V}}_{\mathbf{G}}$ of non-ordered pairs of vertices $(v,w)$ with $v\ne w$ that are called edges.
- (ii)
- A mapping $\mathbf{f}:\mathbf{G}\to \mathbf{H}$ is a mapping $\mathbf{f}:{\mathbf{V}}_{\mathbf{G}}\to {\mathbf{V}}_{\mathbf{H}}$ such that for any edge $(v,w)\in {\mathbf{E}}_{\mathbf{G}}$ we have either $\mathbf{f}\left(v\right)=\mathbf{f}\left(w\right)$ or $(\mathbf{f}\left(v\right),\mathbf{f}\left(w\right))\in {\mathbf{E}}_{\mathbf{H}}$.

**Definition**

**9.**

- (i)
- A coloring of a graph $\mathbf{G}=({\mathbf{V}}_{\mathbf{G}},{\mathbf{E}}_{\mathbf{G}})$ is given by an assignment of a color to each vertex $v\in {\mathbf{V}}_{\mathbf{G}}$. A coloring that uses k colors is called k-coloring. We denote by $(\mathbf{G},\phi )$ a graph $\mathbf{G}$ with a coloring function $\phi :{\mathbf{V}}_{\mathbf{G}}\to \mathbb{N}$.
- (ii)
- Let $(G,\phi )$ and $(\mathbf{H},\psi )$ be two colored graphs. A digraph mapping $\mathbf{f}:\mathbf{G}\to \mathbf{H}$ is a morphism of colored graphs if $\psi \left(f\right(v\left)\right)=\phi \left(v\right)$ for any vertex $v\in {\mathbf{V}}_{\mathbf{G}}$.

**Proposition**

**6.**

**Definition**

**10**

**Remark**

**1.**

**Example**

**6.**

**Theorem**

**7.**

**Definition**

**11.**

**Definition**

**12.**

**Lemma**

**1.**

**Theorem**

**8.**

**Proof.**

**Example**

**7.**

**Remark**

**2.**

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Muranov, Y.V.; Szczepkowska, A.
On Path Homology of Vertex Colored (Di)Graphs. *Symmetry* **2020**, *12*, 965.
https://doi.org/10.3390/sym12060965

**AMA Style**

Muranov YV, Szczepkowska A.
On Path Homology of Vertex Colored (Di)Graphs. *Symmetry*. 2020; 12(6):965.
https://doi.org/10.3390/sym12060965

**Chicago/Turabian Style**

Muranov, Yuri V., and Anna Szczepkowska.
2020. "On Path Homology of Vertex Colored (Di)Graphs" *Symmetry* 12, no. 6: 965.
https://doi.org/10.3390/sym12060965