(Lack of) Model Structures on the Category of Graphs

TL;DR

This paper investigates the possibility of defining model structures on the category of finite graphs with -homotopy equivalences, concluding that such structures do not exist under common conditions, highlighting limitations in this categorical framework.

Contribution

It proves the non-existence of a Strm-Hurewicz type model structure on finite graphs with -homotopy equivalences, revealing fundamental constraints in graph homotopy theory.

Findings

No model structure similar to Strm-Hurewicz exists for finite graphs with -homotopy.

The category of graphs with -homotopy equivalences lacks a model structure when cofibrations are subclasses of graph inclusions.

This limitation impacts the development of homotopical methods in graph theory.

Abstract

In this article, we study model structures on the category of finite graphs with $\times$-homotopy equivalences as the weak equivalences. We show that there does not exist an analogue of Str\o{}m-Hurewicz model structure on this category of graphs. More interestingly, we show that this category of graphs with $\times$-homotopy equivalences does not have a model structure whenever the class of cofibrations is a subclass of graph inclusions.

Equations

$$\chi (G)\ge \chi (T)+\mathrm{conn}(\mathrm{Hom}(T,G))+1.$$

$$V(G/f(A))=V(G)/V(f(A)),E(G/f(A))=\{\{[x],[y]\}\subseteq V(G)/V(f(A))|xy\in E(G)\}.$$

$$C=\Big{(}A\displaystyle\coprod_{x\in A^{\prime}}C_{5}^{x}\Big{)}/\sim$$

$$dis{t}_G(G_1,G_2)=\min \{dis{t}_G(u,v):u\in V(G_1),v\in V(G_2)\}.$$

Full text

Keywords: Model structures, Category of finite graphs, $\times$-homotopy equivalences.

MSC 2010: 55P99; 05C15.

1. Introduction

Let $\mathcal{G}$ denote the category whose objects are finite undirected graphs without multiple edges and morphisms are edge preserving functions on vertices. The class of $\times$-homotopy equivalences in $\mathcal{G}$ defines a class of weak equivalences on the category of graphs, and we denote this category with weak equivalences as $(\mathcal{G},\times)$. These equivalences were defined by Dochtermann in [3] while extending the work of Lovász, Babson, Kozlov on Hom complexes of graphs.

Hom complexes are of interest as their connectivity often gives a lower bound on the chromatic number of a graph. A graph $T$ is a test graph if the following inequality holds for every graph $G$,

[TABLE]

Here, ${\rm Hom}(T,G)$ denotes the Hom complex of graphs $T$ and $G$; and ${\rm conn}(X)$ is the smallest integer $n$, for which there exists a map from the $(n+1)$-sphere $S^{n+1}\to X$ which cannot be extended to a map from the $(n+2)$-disk $D^{n+2}\to X$. In particular, complete graphs, cycle graphs and complete bipartite graphs are all examples of test graphs.

In general, the problem of computing Hom complexes can have high complexity [2]. Dochtermann [3] showed that any graph map $G\to H$ is a $\times$-homotopy equivalence if and only if the induced map on Hom complexes, ${\rm Hom}(T,G)\to{\rm Hom}(T,H)$, is a homotopy equivalence for every graph $T$. Our main motivation for the present work is to be able to replace graphs with $\times$-homotopy equivalent graphs whose Hom complexes would be easier to compute.

Our goal is then to be able to write a graph as a pushout of smaller subgraphs such that this pushout is mapped to a homotopy pushout in spaces. In [6], we noted that given any graph $T$, the functor ${\rm Hom}(T,\underline{\ \ })$ maps the double mapping cylinder, $B\sqcup_{f}(A\times I_{m})\sqcup_{g}C$, to ${\rm Hom}(T,B)\sqcup_{f_{T}}({\rm Hom}(T,A)\times I)\sqcup_{g_{T}}{\rm Hom}(T,C)$. In topological spaces, when the maps in the pushout diagram are Hurewicz cofibrations [1, Proposition 6.2.6], the pushout is a homotopy pushout, that is, a pushout which is preserved (up to a weak equivalence) under weakly equivalent diagrams. However, there is no known criterion in graphs to recognise when a given graph is $\times$-homotopic to a double mapping cylinder of smaller graphs. Further, as explained in [6] the double mapping cylinder is not a correct notion of homotopy pushout in ($\mathcal{G},\times$).

A model structure on $(\mathcal{G},\times)$ will allow us to define the notion of homotopy pushout and will give a criterion to recognise when a pushout is a homotopy pushout (that is, pushouts where the maps are cofibrations). In order to resolve these questions, we require a model structure on $(\mathcal{G},\times)$ with the property that the ${\rm Hom}(T,\underline{\ \ })$ functor will map homotopy pushouts in the category of graphs to homotopy pushouts in the category of spaces. The category of infinite graphs is known to have model structures, albeit with different choices of weak equivalences. Droz constructed different model structures on the category of graphs for which the homotopy type of a graph was the set of its connected components [4, Theorem 4.2], its furbished part [4, Theorem 4.4], its corresponding core graph [4, Theorem 4.13]. Matsushita used the usual model structure on $Top$ to construct a model structure on the category of graphs in [7] and showed that it is Quillen equivalent to the category of $\mathbb{Z}_{2}$-spaces. The class of weak equivalences considered in [7] is the class of maps that induce a $\mathbb{Z}_{2}$-homotopy equivalence on the box complex of graphs (which for a graph $G$, is a simplicial complex homotopy equivalent to ${\rm Hom}(K_{2},G)$). The class of $\times$-homotopy equivalences is a proper subclass of this class of weak equivalences.

Since the notion of $\times$-homotopy equivalence resembles homotopy equivalence from Top, the most natural model structure to expect on $(\mathcal{G},\times)$ is the analogue of the Strøm-Hurewicz model structure on topological spaces. In [8], Strøm showed the existence of a model structure on Top with closed Hurewicz cofibrations, Hurewicz fibrations and homotopy equivalences as the class of cofibrations, fibrations and weak equivalences, respectively. We show that there is no analogue of the Strøm-Hurewicz model structure on $(\mathcal{G},\times)$ (cf. Theorem 2.11).

Based on our motivation to break down a graph as a homotopy pushout of smaller graphs, we want the cofibrations in our model structure on $(\mathcal{G},\times)$ to be a subclass of inclusions (up to isomorphisms) in $\mathcal{G}$. In Remark 3.2, we point out that in order to satisfy the required axioms, we need cofibrations to be induced inclusions. Further, we note that (cf. Remark 3.3) the class of cofibrations for any model structure on $(\mathcal{G},\times)$ cannot be the class of all induced inclusions. In Proposition 3.6, we show that if cofibrations are any subclass of induced inclusions in graphs, then every acyclic cofibration has to be a composition of unfolds. This result then allows us to prove Theorem 4.5 which states that there is no model structure on $(\mathcal{G},\times)$ even if we restrict our class of cofibrations to a subclass of induced inclusions. The proof is based on the lack of compatibility between cofibrations and fibrations in the given scenario.

We restrict ourselves to the category of finite graphs, since to reach $\times$-homotopy equivalent graphs we only need a finite number of folds and unfolds. In the case of infinite graphs, there are other ways to show that $\times$-homotopy equivalences do not give a good class of weak equivalences to form a model structure. In both the scenarios, whether we consider the category of finite or infinite graphs, the main obstruction to having a model structure seems to be that acyclic cofibrations will be a composition of unfolds. We study whether we can instead have a cofibration category structure on $(\mathcal{G},\times)$ in a forthcoming article.

2. Homotopy Extensions in Graphs

A graph $G$ is a pair of sets $(V(G),E(G))$. The elements of $V(G)$ are called vertices of $G$, and the elements of $E(G)$, which are two-element subsets of $V(G)$, are called edges of $G$. These elements need not be distinct, that is, $\{y,y\}$ can also be an edge and this makes $y$ into a looped vertex. We say that two vertices $x$, $y$ are adjacent if $\{x,y\}\in E(G)$ and denote the edge by $xy$. A simple graph is a graph without any looped vertex and multiple edges. A complete graph is a simple graph where any two distinct vertices are adjacent to each other. A finite graph is a graph with a finite set of vertices. A graph map between two graphs $f:G\to H$ is a function from $V(G)$ to $V(H)$ such that if $xy$ is an edge in $G$, then $f(x)f(y)$ is an edge in $H$.

Definition 2.1**.**

Let $G,H\in\mathcal{G}$ be two graphs. The (categorical) product of $G$ and $H$ is defined to be the graph $G\times H$ whose vertex set is the cartesian product, $V(G)\times V(H)$, and $(g,h)(g^{\prime},h^{\prime})\in E(G\times H)$ whenever $gg^{\prime}\in E(G)$ and $\ hh^{\prime}\in E(H)$.

Let $G$ be a graph and $v\in V(G)$ be a vertex of $G$. The neighbourhood set of $v$ in $G$ is the set $\{x\in V(G):vx\in E(G)\}$, and is denoted by ${\rm N}_{G}(v)$. We drop the subscript $G$ in ${\rm N}_{G}(v)$, and we write ${\rm N}(v)$ whenever the choice of graph is clear.

Definition 2.2**.**

A vertex $v$ of a graph $G$ is said to fold to a vertex $v^{\prime}\in V(G)$ if every neighbour of $v$ is also a neighbour of $v^{\prime}$, that is, ${\rm N}(v)\subseteq{\rm N}(v^{\prime})$. We note that the map $f:G\to G-v$ that maps each vertex (other than $v$) of $G$ to itself, and $v$ to $v^{\prime}$ is a graph map. In such a case, we call $f$ a fold map that folds $G$ to $G-v$. If a graph folds to a single looped vertex via a sequence of fold maps, then it is called a contractible graph. Dual to a fold map, an unfold is defined to be the inclusion map $A\to A\cup\{v\}$ such that ${\rm N}_{A\cup\{v\}}(v)\subseteq{\rm N}_{A}(v^{\prime})$ for some $v^{\prime}\in V(A)$, where $v\notin V(A)$.

Definition 2.3**.**

A graph is stiff if there is no fold in that graph. By a stiff subgraph of a graph $G$ we mean a subgraph of $G$ which is a stiff graph and can be obtained from $G$ via a sequence of folds. It is easy to observe that if $S\subset G$ is a stiff subgraph of $G$ and $S$ is simple, then $G$ is also a simple graph.

For $n\in\mathbb{N}$, let $I_{n}$ be the graph with vertex set $\{0,1,\dots,n\}$ and edge set $\{ij:|i-j|\leq 1\}$. We note that the graph $I_{n}$ folds down to a single looped vertex and hence is contractible for any $n\in\mathbb{N}$.

Definition 2.4**.**

[3, Definition 4.3] Two graph maps $f,g:A\to B$ are said to be $\times$-homotopic if for some $n\in\mathbb{N}$, there exists a graph map $F:A\times I_{n}\to B$ such that $F|_{A\times\{0\}}=f$ and $F|_{A\times\{n\}}=g$. Two $\times$-homotopic maps are denoted as $f\simeq_{\times}g$. A graph map $f:A\to B$ is a $\times$-homotopy equivalence, if there exists a graph map $g:B\to A$ such that $gf\simeq_{\times}1_{A}$ and $fg\simeq_{\times}1_{B}$, where $1_{X}$ denotes the identity map on $X$.

It is easy to see that the fold and unfold maps are $\times$-homotopy equivalences. Thus every graph is $\times$-homotopy equivalent to its stiff subgraph. Moreover, if $f:A\to B$ is a $\times$-homotopy equivalence between stiff graphs $A$ and $B$, then $f$ is an isomorphism [3, Thm 6.6].

Lemma 2.5**.**

*Let $f:G\rightarrow H$ be a $\times$-homotopy equivalence of graphs. Let $G^{\prime}$ and $H^{\prime}$ be stiff subgraphs of $G$ and $H$ respectively. Then there exists a commuting diagram as shown below, where $f^{\prime}:G^{\prime}\rightarrow H^{\prime}$ an isomorphism and the vertical maps are a sequence of folds to their respective stiff subgraphs.

\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{H^{\prime}} *

Proof.

The graph map $f^{\prime}:G^{\prime}\to H^{\prime}$ is defined as the composition of the sequence of unfolds from $G^{\prime}$ to $G$, $f:G\to H$ and sequence of folds from $H$ to $H^{\prime}$. Then $f^{\prime}:G^{\prime}\to H^{\prime}$ is a $\times$-homotopy equivalence between stiff graphs and by [3, Thm 6.6] $f^{\prime}$ is an isomorphism. ∎

Let $Top$ denote the category of topological spaces with continuous functions. Recall that if $A$ and $B$ are topological spaces, then a continuous map $i:A\to B$ that has the homotopy extension property  with respect to every $Z\in Top$ is called a Hurewicz cofibration.

Dual to the notion of homotopy extension property, there is homotopy lifting property;   and a continuous map $p:X\to Y$ in $Top$ with the homotopy lifting property  with respect to every space $Z\in Top$ is called a Hurewicz fibration. Before we proceed further, we give the definition of a model category.

Definition 2.6**.**

[5] A category $\mathcal{C}$ with three distinguished classes of morphisms - cofibrations, fibrations and weak equivalences is said to be a model category if it satisfies the following axioms:

1. (1)

(Bicompleteness) All the finite limits and finite colimits exists in the category. 2. (2)

(Retract) If $f$ and $g$ are morphisms of the category such that $f$ is a retract of $g$ and $g$ is a weak equivalence, cofibration, or fibration, then so is $f$. 3. (3)

(2 out of 3) If $f$ and $g$ are morphisms of the category such that $gf$ is defined and two of $f,\ g$ and $gf$ are weak equivalences, then so is the third. 4. (4)

(Lifting) Acyclic cofibrations111A morphism that is both a weak equivalence and a cofibration is called an acyclic cofibration. have the left lifting property with respect to fibrations, and cofibrations have the left lifting property with respect to acyclic fibrations222A morphism that is both a weak equivalence and a fibration is called an acyclic fibration.. 5. (5)

(Factorization) Any morphism of the category can be factored as a cofibration followed by an acyclic fibration, and as acyclic cofibration followed by a fibration.

By a model structure on the category we mean a choice of class of cofibrations, fibrations and weak equivalences satisfying the above properties.

Strøm showed that $Top$ has a model structure (cf. [8, Theorem 11]) with closed Hurewicz cofibrations, Hurewicz fibrations and homotopy equivalences as the class of cofibrations, fibrations and weak equivalences respectively. We now study analogous constructions in the category of graphs, $\mathcal{G}$.

Definition 2.7**.**

Let $A$ and $B$ be graphs and the maps $j_{0}:A\to A\times I_{n}$, $k_{0}:B\to B\times I_{n}$ denote graph maps that take an element $x$ to $(x,0)$, for $x$ in $A$ and $B$, respectively. A graph map $i:A\to B$ is said to have the $\times$-homotopy extension property if for any graph $Z\in\mathcal{G}$ with graph maps $f:A\to Z,g:B\to Z$, and a $\times$-homotopy $F:A\times I_{n}\to Z$ satisfying $k_{0}i=(i\times 1_{I_{n}})j_{0},f=Fj_{0}$ and $gi=f$, there exists a $\times$-homotopy $G:B\times I_{n}\to Z$ that extends $g$, that is, $G(i\times 1_{I_{n}})=F$ and $Gk_{0}=g$. Equivalently, we require that the dotted arrow should exist for every commutative diagram as in Figure 1.

Similarly, a graph map $p:X\to Y$ is said to have the $\times$-homotopy lifting property if for any graph $A$ and a commutative diagram as in Figure 2, there exists a diagonal $G:A\times I_{n}\to X$ such that $Gi_{0}=f$ and $pG=g$, where $i_{0}(a)=(a,0)$.

By a Strøm-Hurewicz type model structure on $(\mathcal{G},\times)$, we mean a model structure on $\mathcal{G}$ where, cofibrations have the $\times$-homotopy extension property, fibrations have the $\times$-homotopy lifting property, and weak equivalences are $\times$-homotopy equivalences.

Let $A,B\in\mathcal{G}$ be two graphs and $i:A\to B$ be a graph map. Then the graph $A$ is called a retract of $B$ if there exists a graph map $r:B\to A$ such that $ri=1_{A}$. We call this map $r:B\to A$, a retraction of $B$ onto $A$. We note that if $A$ is a retract of $B$, then $A$ is a subgraph, up to a relabelling of vertices of $A$, of $B$.

Let $G\in\mathcal{G}$ be a graph, and $f:A\to G$ be a graph map. The quotient of $G$ by $f(A)$, denoted by $G/f(A)$ is the graph defined as

[TABLE]

Let $i:A\to B$ be a graph map. For $n\in\mathbb{N}$, let $(A\times I_{n})\bigsqcup\limits_{i}B$ be the quotient graph such that V\Big{(}(A\times I_{n})\bigsqcup\limits_{i}B\Big{)}=\Big{(}V(A\times I_{n})\cup V(B)\Big{)}/i(a)\sim(a,0) and [x][y]\in E\Big{(}(A\times I_{n})\bigsqcup\limits_{i}B\Big{)} if and only if there exist $x^{\prime}\in[x],y^{\prime}\in[y]$ such that $x^{\prime}y^{\prime}\in E(A\times I_{n})\cup E(B)$.

Let $I$ denote the unit interval $[0,1]$ in $Top$. We recall that a closed continuous map $i:A\to B$ is a closed Hurewicz cofibration if and only if $(A\times I)\cup(B\times\{0\})$ is a retract of $B\times I$. The analog in graphs is true as well.

Theorem 2.8**.**

A graph map $i:A\to B$ has the $\times$-homotopy extension property if and only if $(A\times I_{n})\bigsqcup\limits_{i}B$ is a retract of $B\times I_{n}$, for every $n\in\mathbb{N}$.

Proof.

Let $i:A\to B$ have the $\times$-homotopy extension property, that is, for any graph $Z$ and maps $f,F$, and $g$ as in Figure 1, there exists a $\times$-homotopy extension $G$. Let $n\in\mathbb{N}$ be fixed, choose $Z=(A\times I_{n})\bigsqcup\limits_{i}B$ together with $F:A\times I_{n}\to Z$ that sends any $(a,j)\in A\times I_{n}$ to $(a,j)\in(A\times I_{n})\cup B$, $f=F|_{A\times I_{0}}$ and $g:B\to Z$ as $g(b)=b$ for all $b\in B$. By assumption, there exists $G:B\times I_{n}\to(A\times I_{n})\bigsqcup\limits_{i}B$ such that $G|_{(A\times I_{n})\bigsqcup\limits_{i}B}=1_{(A\times I_{n})\bigsqcup\limits_{i}B}$. Therefore, $G$ is a retraction of $B\times I_{n}$ onto $(A\times I_{n})\bigsqcup\limits_{i}B$.

Next, let $Z\in\mathcal{G}$ be any graph with $f:A\to Z$, $g:B\to Z$ and $F:A\times I_{n}\to Z$ be such that $F(a,0)=f(a)$. Let $r:B\times I_{n}\to(A\times I_{n})\bigsqcup\limits_{i}B$ be a retraction. Define the map $H:(A\times I_{n})\bigsqcup\limits_{i}B\to Z$ as $H(a,t)=F(a,t)$ for all $a\in A$, $t\in I_{n}$, and $H(b)=g(b)$ for all $b\in B$. Then $H(a,0)=F(a,0)=f(a)=gi(a)$. Since $[(a,0)]=[i(a)]$ in $(A\times I_{n})\bigsqcup\limits_{i}B$, $Hi(a)=gi(a)$. Since $i(a)\in B$, $H$ is a well defined map. Also, by construction it is a graph map. We define the map $G:B\times I_{n}\to Z$ as $G=Hr$, then $G$ is a $\times$-homotopy that extends $g:B\to Z$. ∎

Let $i:A\to B$ be a graph map. It is easy to see that if $i$ is an isomorphism or $B$ is a disjoint union of $A$ and $B-Im(i)$, then $i$ has the $\times$-homotopy extension property. Unlike in $Top$, the converse is also true in $\mathcal{G}$.

Lemma 2.9**.**

Let $i:A\to B$ be a graph map such that $i$ is not an isomorphism and $B\neq A\sqcup(B-Im(i))$. Then there does not exist a retraction $r:B\times I_{n}\to(A\times I_{n})\bigsqcup\limits_{i}B$.

Proof.

Since $i$ is not an isomorphism and $B\neq A\sqcup(B-A)$, there exist $a\in A$, $b\in B\setminus i(A)$ such that $i(a)$ is adjacent to $b$ in $B$. Suppose there exists a retraction $r:B\times I_{n}\to(A\times I_{n})\bigsqcup\limits_{i}B$. Then $(b,0)$ is adjacent to $(i(a),1)$ in $B\times I_{n}$. Since $r$ is a graph map, $r(b,0)$ is adjacent to $r(i(a),1)$ in $(A\times I_{n})\bigsqcup\limits_{i}B$. However, $rj=1_{(A\times I_{n})\bigsqcup\limits_{i}B}$ where $j:(A\times I_{n})\bigsqcup\limits_{i}B\to B\times I_{n}$ is an inclusion. Therefore $r(b,0)=(b,0)$ and $r(i(a),1)=(i(a),1)$. But $(b,0)$ is not adjacent to $(i(a),1)$ in $(A\times I_{n})\bigsqcup\limits_{i}B$. Hence, there does not exist any $r:B\times I_{n}\to(A\times I_{n})\bigsqcup\limits_{i}B$ such that $rj=1_{(A\times I_{n})\bigsqcup\limits_{i}B}$. ∎

If there exists a model structure on a category where two of the three special classes of morphisms are chosen to be full morphism class of the category, and the third one is chosen to be isomorphisms in that category, then such a model structure is often called a trivial model structure on the category. Droz has shown that:

Theorem 2.10** (Droz, [4, Theorem 4.1]).**

There exists a trivial model structure on the category of graphs, $\mathcal{G}$, with the class of cofibrations as the isomorphisms of $\mathcal{G}$ .

Since we are interested in finding the cofibrant replacement of a graph with respect to $\times$-homotopy equivalences as the class of weak equivalences, this structure is not useful for us. Let the class of cofibrations $A\to B$ be isomorphisms or of the form $A\to A\sqcup B^{\prime}$ and weak equivalences be $\times$-homotopy equivalences. Then regardless of our class of fibrations, it is not possible to factor graph maps $X\to Y$, where both $X$ and $Y$ are connected, as a cofibration followed by an acyclic fibration unless the map itself is a $\times$-homotopy equivalence. Thus we have proved the following theorem.

Theorem 2.11**.**

There does not exist a Strøm-Hurewicz type model structure on $(\mathcal{G},\times)$.

This theorem is a special case of Theorem 4.5 which we prove later in the paper. We include the statement for 2.11 here since the above argument is straightforward and does not need the techniques used in the proof of Theorem 4.5.

In the next section, we give a characterization of the class of cofibrations given any model structure on $(\mathcal{G},\times)$ under mild assumptions.

3. Criteria for cofibrations in $(\mathcal{G},\times)$

In any model category, the class of fibrations and the class of cofibrations determine each other completely [5, Proposition 3.13] when we fix the class of weak equivalences. In this section, we analyse the possible classes of maps which would give a compatible choice of cofibrations. In line with our original motivation of replacing graphs with smaller graphs, the only restriction we impose on the class of cofibrations is that they be inclusions.

We note that in a model category [5, Proposition 3.14], an acyclic cofibration should be preserved under the cobase change along any morphism. Since a $\times$-homotopy equivalence is a composition of folds, unfolds and isomorphisms, we next analyse the behaviour of such maps and inclusions under the cobase change along an arbitrary graph map.

Lemma 3.1**.**

Let $i:A\to B$ be a graph map. If $i$ is an isomorphism, inclusion, induced subgraph inclusion or an unfold then so is the cobase change of $i$ along any graph map.

Proof.

Let $C\in\mathcal{G}$ be any graph and $f:A\to C$ be a graph map. Let $G\in\mathcal{G}$ be the pushout of $\{f,i\}$ as shown in Figure 3.

It is an easy observation that the cobase change of an isomorphism along any graph map is also an isomorphism. Since set inclusions are preserved under pushouts, the cobase change of a graph inclusion along a graph map will be a graph inclusion.

Next, let $i$ be an induced subgraph inclusion, that is, for every $ab\in E(B)$, if $a,b\in V(A)$, then $ab\in E(A)$. As noted earlier $i^{\prime}$ is an inclusion and therefore it is enough to show that if $[x][y]=i^{\prime}(x)i^{\prime}(y)\in E(G)$ for some $x,y\in V(C)$, then $xy\in E(C)$. Again if one of $x$ or $y$ does not belong to $f(A)$, then $[x][y]\in E(G)$ implies $xy\in E(C)$. So, let $x,y\in f(A)$, that is, for some $a,b\in V(A)$, $f(a)=x$ and $f(b)=y$. Since $[f(a)]=[x]$ is adjacent to $[y]=[f(b)]$, either $i(a)i(b)\in E(B)$ or $xy\in E(C)$. Assume that $i(a)i(b)\in E(B)$ then $i$ being induced subgraph inclusion implies that $ab\in E(A)$. Given that $f$ is a graph map, $ab\in E(A)$ gives $f(a)f(b)=xy\in E(C)$.

Let $B=A\cup\{b\}$ and there exist $a\in V(A)$ such that $b$ folds to $a$ in $B$, that is, ${\rm N}_{B}(b)\subseteq{\rm N}_{B}(a)={\rm N}_{A}(a)$ (or ${\rm N}_{A}(a)\cup\{b\}$ if $b$ is looped). Then the pushout graph is $G$ with $V(G)=V(C)\cup\{b\}$ with vertices $x\in V(A)$ identified with $f(x)\in V(C)$. Since an unfold graph map is an induced inclusion, we know that $i^{\prime}$ is an induced subgraph inclusion. Further, $G$ is a pushout implies that the neighbours of $a$ in $A$ will get identified with neighbours of $f(a)$ in $C$. Therefore, ${\rm N}_{G}([a])={\rm N}_{G}([f(a])={\rm N}_{C}(f(a))$. Note that ${\rm N}_{G}([b])=f^{\prime}({\rm N}_{B}(b))$. Since ${\rm N}_{B}(b)\subseteq{\rm N}_{A}(a)$ (or $N_{A}(a)\cup\{b\}$ if $b$ is looped) and $f^{\prime}$ is a graph map, we get that ${\rm N}_{G}([b])=f^{\prime}({\rm N}_{B}(b))\subseteq f^{\prime}({\rm N}_{A}(a))$ (or $f^{\prime}(N_{A}(a)\cup\{b\}$ if $b$ is looped). Now $f^{\prime}({\rm N}_{A}(a))=f({\rm N}_{A}(a))\subseteq{\rm N}_{C}(f(a))={\rm N}_{G}([f(a)])$ (or $f^{\prime}({\rm N}_{A}(a))\cup\{b\}\subseteq{\rm N}_{C}(f(a))\cup\{b\}={\rm N}_{G}([f(a)]$ if $b$ is looped). Hence $b$ folds to $[f(a)]$ in $G$. ∎

Remark 3.2**.**

We first observe that every cofibration must be an induced inclusion. Given a $\times$-homotopy equivalence $i:A\to B$ which is not an induced inclusion, we can always construct a graph map $g:A\to\widetilde{C}_{4}$ (where the graph $\widetilde{C}_{4}$ denotes the 4-cycle having loop at each vertex) along which the cobase change of $i$ will not be a $\times$-homotopy equivalence. Since $i$ is not induced, there exist $a_{1},a_{2}\in V(A)$ such that $a_{1}a_{2}\notin E(A)$ and $a_{1}a_{2}\in E(B)$.

Consider the graph map $g:A\to\widetilde{C}_{4}$ as in Figure 4 that sends $a_{1}$ to 1, $a_{2}$ to 3, and rest of $A$ to $2$. Let $G$ be the pushout graph of $\{i,g\}$ and $h$ be the cobase change map of $g$ along $\alpha$.

Suppose that the cobase change map $\alpha$ of $i$ along $g$ is a $\times$-homotopy equivalence. Since $\widetilde{C}_{4}$ is stiff, $G$ has a subgraph isomorphic to $\widetilde{C}_{4}$, say, $X$ with $V(X)=\{x_{1},x_{2},x_{3},x_{4}\}$ (see Figure 4), to which $G$ folds down. By Lemma 2.5, we get a commuting triangle as in Figure 4, where $\cong$ denotes isomorphism. Now $a_{1}a_{2}\in E(B)$ implies that $h(a_{1})h(a_{2})\in E(G)$ and therefore, $\alpha(1)\alpha(3)\in E(G)$. For every $1\leq j\leq 4$, $\alpha(j)$ folds in $G$ to some vertex of $X$. Let $\alpha(1)$ fold to $x_{1}$. Since $\alpha(j)$ is adjacent to $\alpha(1)$, for each $j=2,3,4$, it must fold to a vertex in $\{x_{1},x_{2},x_{4}\}=V(X)\setminus\{x_{3}\}$. But this sequence of folds then cannot commute with the isomorphism as $1$ and $3$ are not adjacent in $\widetilde{C}_{4}$ . This contradicts our assumption that $B$ is $\times$-homotopy equivalent to $A$.

A short and elegant proof of the above argument based on the referee’s suggestion can be obtained by using the notion of Hom complexes and [3, Lemma 6.5].

Remark 3.3**.**

Next, we observe that in $(\mathcal{G},\times)$ not all induced inclusions that are $\times$-homotopy equivalences are preserved under the cobase change along every graph map.

For instance, let $C$ be the graph on the left bottom side in Figure 5. Consider the graph $A$ (top left) and $B$ (top right) as the induced subgraphs of $C$ (as shown in Figure 5) on the vertex sets $\{a,b,c,p\}$ and $\{a,b,c\}$ respectively. Let $f:A\to B$ be the fold map that sends $p$ to $a$ and $g:A\to C$ be the inclusion. We note that $g$ is a $\times$-homotopy equivalence.

Then the pushout of $\{f,g\}$ is isomorphic to $K_{4}$, while $C\simeq_{\times}K_{3}$. Therefore the cobase change along $f$ of the induced inclusion map $g$ which is also a $\times$-homotopy equivalence is not a $\times$-homotopy equivalence. This example indicates that if $g$ is a composition of folds and unfolds then this property may not be preserved under the cobase change along some graph map even if $g$ is an induced inclusion.

Based on the above remarks, we know that the class of cofibrations should be a subclass of induced inclusions. Further, it is clear that it has to be a proper subclass of induced inclusions. We now establish that regardless of which subclass of induced inclusions we choose as the class of cofibrations, the acyclic cofibrations have to be a subclass of unfolds.

3.1. A necessary condition for acyclic cofibrations

Definition 3.4**.**

Let $A$ be a subgraph of $B$. For some vertex $v\in V(B)$, a fold map $f:B\to B-v$ is called a relative fold in $B$ with respect to $A$, if $A$ is a subgraph of $B-v$.

Lemma 3.5**.**

Let $i:A\to B$ be an induced subgraph inclusion and a $\times$-homotopy equivalence. Let $f:B\to B-v$ be a relative fold in $B$ with respect to $A$, $j:A\to B-v$ be the subgraph inclusion and $\alpha:A\to X$ be any graph map. Then the cobase change of the inclusion map $j:A\to B-v$ along $\alpha$ is a $\times$-homotopy equivalence if and only if the cobase change of $i$ along $\alpha$ is.

Proof.

Let $g:B-v\to B$ be the unfold map corresponding to the fold map $f:B\to B-v$. Suppose the cobase change of $j:A\to B-v$ along $\alpha$ is a $\times$-homotopy equivalence for every graph map $\alpha$ defined on $A$. By Lemma 3.1, an unfold map is preserved under the cobase change along any graph map, therefore any cobase change of $g$ is an unfold map. In particular, any cobase change of $g$ is a $\times$-homotopy equivalence, and hence any cobase change of the composite map $gj=i$ is a $\times$-homotopy equivalence.

Now assume that the cobase change of $i:A\to B$ along any graph map $\alpha:A\to X$ is a $\times$-homotopy equivalence. We show that the cobase change of $j:A\to B-v$ along any graph map $\alpha:A\to X$ is also a $\times$-homotopy equivalence. Let $G_{1}$ be the pushout of $\{j,\alpha\}$, with cobase change maps $j^{\prime}:X\to G_{1}$, and $\alpha^{\prime}:B-v\to G_{1}$, as shown in Figure 6.

Let $G_{2}$ be the pushout of $\{g,\alpha^{\prime}\}$ along with cobase change maps $g^{\prime}:G_{1}\to G_{2}$,and $\alpha^{\prime\prime}:B\to G_{2}$. We note that the pushout of $\{\alpha,gj\}$ is isomorphic to $G_{2}$, as $j$ and $g$ are inclusions. Then $g^{\prime}j^{\prime}$ is a $\times$-homotopy equivalence by assumption, and $g^{\prime}$ is a $\times$-homotopy equivalence by Lemma 3.1. By 2 out of 3 property [3, Lemma 5.5] of $\times$-homotopy equivalences, this shows that $j^{\prime}$ is a $\times$-homotopy equivalence. Thus $j$ is also preserved under the cobase change along any graph map $\alpha:A\to X$. ∎

Proposition 3.6**.**

Let $i:A\to B$ be an induced subgraph inclusion map and a $\times$-homotopy equivalence such that $|V(A)|<|V(B)|$. If $B$ does not fold to $A$ via some sequence of relative folds, then there exists a graph map along which the cobase change of $i$ is not a $\times$-homotopy equivalence.

Proof.

Suppose there exists a fold of a vertex $x$ in $B$ relative to $A$. In view of Lemma 3.5, the cobase change of $i:A\to B$ along $\alpha$ is a $\times$-homotopy equivalence  for every map $\alpha$ if and only if the cobase change of $A\to B-x$ is. Therefore without any loss of generality, we assume that for $i:A\to B$, there is no relative fold in $B$ with respect to $A$.

In view of [3, Proposition 6.6], given two graphs $G$ and $H$ with stiff subgraphs $S_{G}$ and $S_{H}$ respectively, $S_{G}\simeq_{\times}S_{H}$ if and only if $S_{G}$ is isomorphic to $S_{H}$. Since every graph is $\times$-homotopy equivalent to its stiff subgraph, any two $\times$-homotopy equivalent graphs must have isomorphic stiff subgraphs. Therefore $|V(A)|<|V(B)|$ implies that $B$ is not stiff. Thus there exists a $v\in V(B)$ that folds. If $v\in V(B)-V(A)$, then $A\subseteq B-v$ implies a relative fold in $B$ with respect to $A$ which contradicts our assumption. Thus any vertex that folds in $B$ is a vertex of $A$.

Define $A^{\prime}$ to be the set consisting of vertices of $A$ which fold in $B$. For every $x\in A^{\prime}$, we associate a 5-cycle denoted $C_{5}^{x}$ with vertex set $\{x_{1},x_{2},\dots,x_{5}\}$ and edge set $\{x_{1}x_{2},x_{2}x_{3},x_{3}x_{4},x_{4}x_{5},\\ x_{5}x_{1}\}$. We now define a graph $C$ as follows:

[TABLE]

where ‘$\sim$’ denotes the identification of $x$ with $x_{1}$, for $x\in A^{\prime}\subseteq V(A)$.

Let $f:A\to C$ be the subgraph inclusion. Consider the pushout object $G$ of $\{i,f\}$ and the cobase change graph map $i^{\prime}:C\to G$ of $i$ along $f$. If the cobase change $i^{\prime}$ of $i:A\to B$ along $f$ is a $\times$-homotopy equivalence, then the stiff subgraphs of $C$ and $G$ are isomorphic. In particular, cardinality of vertex sets of stiff subgraphs of $C$ and $G$ are equal. We note that $|V(A)|<|V(B)|$, and $A^{\prime}\subset V(A)$ implies that $|V(C)|<|V(G)|$.

If $G$ is not stiff, then there exists a vertex $a\in V(G)$ that folds. We note that a 5-cycle $C_{5}$ is a stiff graph, and any of these cycles might have gained a loop on $x_{1}$’s where it is identified in $G$. However, a 5-cycle with a loop on any of its vertices is also a stiff graph. Therefore, no vertex of $\displaystyle\bigcup_{x\in A^{\prime}}\{x_{2},x_{3},x_{4},x_{5}\}$ can fold down to any vertex in $G$. Further, if $x_{1}\in G$ folds to some vertex in $G$ then $x_{2},x_{5}$ must be a neighbour of that vertex. Since ${\rm N}_{G}(x_{2})\cap{\rm N}_{G}(x_{5})=\{x_{1}\}$, this is not possible.

If $a\in G$ which folds is not a vertex of these copies of $C_{5}$ then $a\in V(B)-A^{\prime}$ and ${\rm N}_{G}(a)={\rm N}_{B}(a)$. Since $B$ does not have relative folds with respect to $A$ and the vertices in $B$ which fold are elements of $A^{\prime}$, there does not exist any such $a\in G$. Therefore, the proposition follows. ∎

4. Lack of model structures on $(\mathcal{G},\times)$

[TABLE]

Lemma 4.1**.**

Every unfold map has the left lifting property with respect to  $p_{n}$ for every $n$.

Proof.

Let $i:X\to X\cup x$ be an unfold map, and $f:X\to L_{n}$, $g:X\cup x\to I_{0}$ be any graph maps such that $p_{n}f=gi$ (see Figure 7). Since image of $p_{n}$ is a single looped vertex, $g(x)=g(x^{\prime})$, where $x^{\prime}$ is a vertex from which $x$ is unfolded. Then the map $F:X\cup x\to L_{n}$ defined as $F|_{X}=f$ and $F(x)=f(x^{\prime})$, is a graph map that satisfies $Fi=f$ and $p_{n}F=g$. This implies that every unfold map has left lifting property with respect to  $p_{n}$.

∎

Corollary 4.2**.**

If $\mathcal{M}$ is a model structure on $\mathcal{G}$ whose weak equivalences are $\times$-homotopy equivalences, and every acyclic cofibration is a composition of unfolds, then the graph map $p_{n}$ defined above is an acyclic fibration.

Proof.

Note that $L_{n}$ folds to $I_{0}$, and hence for any $n$, $p_{n}$ is a $\times$-homotopy equivalence. By Lemma 4.1, $p_{n}$ lifts on the right of every unfold. Therefore, for every $n\in\mathbb{N}$, $p_{n}$ will lift against all compositions of unfolds and hence against acyclic cofibrations. Since in a model structure any map which lifts against all acyclic cofibrations is a fibration [5, Prop 3.13] we get that $p_{n}$ is an acyclic fibration. ∎

Definition 4.3**.**

Distance between any two vertices $u,v$ of a graph $G$ is the number of edges in a shortest length path connecting the two vertices, denoted $dist_{G}(u,v)$. For two subgraphs $G_{1},G_{2}$ of $G$, the distance between them, $dist_{G}(G_{1},G_{2})$, is defined as

[TABLE]

Lemma 4.4**.**

Let $i:K_{2}\to C_{n}$ be an inclusion and $n$ odd. Then $i$ does not lift against $p_{t}$ for any natural number $t>n$.

Proof.

Define $g:C_{n}\to I_{0}$ to be the constant graph map. Define $f:K_{2}\to L_{t}$ as $f(1)=t,f(2)=t-1$. Since both $p_{t}$ and $g$ are constant maps, $p_{t}f=gi$. However, $t>n$ implies that given any graph map $F:C_{n}\to L_{t}$ such that $p_{t}F=g$ and $Fi=f$, image of $F$ does not include the vertex [math]. Therefore, $Im(F)$ is a simple subgraph of $L_{t}$ and in particular, is a path graph say $P_{l}$. Thus existence of $F$ implies that there is a map from an odd cycle $C_{n}$ onto a path graph. This is not possible, since a graph map $C_{n}\to P_{l}$ implies that the chromatic number of $C_{n}$ cannot be bigger than the chromatic number of $P_{l}$. ∎

More generally, the above argument can be used to prove that for any graph $B$, if $f:K_{2}\to B$ factors through an odd cycle then it does not lift against $p_{k}$ for some $k$.

Theorem 4.5**.**

If the class of cofibrations is any subclass of induced subgraph inclusions, then there does not exist a model category structure on $(\mathcal{G},\times)$.

Proof.

Suppose $(\mathcal{G},\times)$ has a model category structure with the class of cofibrations $\mathcal{C}$ as a subclass of all induced subgraph inclusions. By Proposition 3.6, the class of acyclic cofibrations has to be a subclass of all compositions of unfolds. Further by Corollary 4.2, we know that in any such model structure, the graph map $p_{n}:L_{n}\to I_{0}$ must be an acyclic fibration.

Consider $f:K_{2}\to C_{n}$, for $n$ odd. Clearly, this is not a $\times$-homotopy equivalence. Further, $f$ is an inclusion by Lemma 4.4, we see that $f$ is not a cofibration. Then $f$ can be factored into a cofibration $K_{2}\to B$ and an acyclic fibration $B\to C_{n}$, with neither of these maps being isomorphisms.

Since any cycle graph $C_{n}$ for $n\neq 4$ is a stiff graph, $B$ has an isomorphic copy of $C_{n}$, say $\widetilde{C_{n}}$, to which the image of $K_{2}$, say $\widetilde{K_{2}}$, in $B$ must fold. If $\widetilde{K_{2}}$ is in $\widetilde{C_{n}}$, then Lemma 4.4 implies $f$ cannot be a cofibration. If $\widetilde{K_{2}}$ is not in $\widetilde{C_{n}}$, then the edge $v_{1}v_{2}$ of $\widetilde{K_{2}}$ will fold down via a sequence of folds to an edge in $\widetilde{C_{n}}$. Let the distance from $v_{1}v_{2}$ to the edge it folds down to in $\widetilde{C_{n}}$ be $r$. Then an argument similar to one used in Lemma 4.4 will show that $K_{2}\to B$ does not lift against $p_{r+2+t}$ for all natural numbers $t>n$. This implies $K_{2}\to C_{n}$ cannot be factorized and hence there is no such model structure on $(\mathcal{G},\times)$ with $\mathcal{C}$ as the cofibrations. ∎

As elucidated earlier, the non-existence of a model structure on $(\mathcal{G},\times)$ where the class of cofibrations are a subclass of inclusions arises from the lack of compatibility with any class of fibrations. Going forward, it is reasonable to look for a structure which allows us to define homotopy pushouts with a less rigid structure. One such structure which can facilitate the existence of homotopy pushouts is that of a cofibration category (for the definition, refer to [9]). In our forthcoming article we explore the existence of a cofibration category structure on $(\mathcal{G},\times)$.

Acknowledgements

We thank the anonymous referee for several suggestions which has improved this article.