Homomorphisms of Cayley graphs and Cycle Double Covers

TL;DR

This paper investigates a conjecture relating Cayley graph homomorphisms to graph flows, demonstrating partial results and implications for cycle double covers, thus advancing understanding in algebraic graph theory.

Contribution

It introduces a strengthened conjecture, provides partial proofs for specific cases, and links the conjecture to cycle double cover existence.

Findings

A natural strengthening of the conjecture does not hold universally.

The original conjecture implies the existence of small oriented cycle double covers.

Partial results support the conjecture for certain subclasses.

Abstract

We study the following conjecture of Matt DeVos: If there is a graph homomorphism from Cayley graph Cay(M, B) to another Cayley graph Cay(M', B') then every graph with an (M, B)-flow has an (M', B')-flow. This conjecture was originally motivated by the flow-tension duality. We show that a natural strengthening of this conjecture does not hold in all cases but we conjecture that it still holds for an interesting subclass of them and we prove a partial result in this direction. We also show that the original conjecture implies the existence of an oriented cycle double cover with a small number of cycles.

Equations

$$\sum _{vw\in E}\phi (vw)-\sum _{uv\in E}\phi (uv)=0.$$

$$\sum _{e\in C^{+}}\tau (e)-\sum _{e\in C^{-}}\tau (e)=0$$

$${\mathcal{H}}_m(x):=\left\{m(a+x)-m(a):a\in M\right\}.$$

$$\Phi :=\sum _{v\in V}{3}^{\mathrm{deg}v}.$$

$${\phi }^{\prime \prime }(e)\in {\mathcal{H}}_h[{\mathcal{H}}_m(\phi (e))]=\left\{h(m(a-x)-m(a)):a\in M\right\}=$$

$$=\left\{h(m(a-x))-h(m(a)):a\in M\right\}={\mathcal{H}}_{h\circ m}(x).\text{qed}$$

$$x⟼g_x$$

$$\mathcal{H}(01)$$

$$\mathcal{H}(10)$$

$$\mathcal{H}(11)$$

$$e\in C_1$$

$$e\in C_2$$

$$e\in C_3$$

$$\psi =\frac{C_1+C_2+C_3}{2}a+\frac{C_1-C_2-C_3}{2}b+\frac{-C_1+C_2-C_3}{2}c+\frac{-C_1-C_2+C_3}{2}d$$

$$\mathcal{H}(1)$$

$$\mathcal{H}(2)$$

$$\mathcal{H}(3)$$

$$e\in C_1$$

$$e\in C_2$$

$$e\in C_3$$

$$\psi =\frac{-C_1+C_2-C_3}{2}a+\frac{C_1-C_2-C_3}{2}b+\frac{C_1+C_2+C_3}{2}c+\frac{-C_1-C_2+C_3}{2}d$$

Full text

1 Introduction

For an abelian group $M$ (all groups in this article are abelian even though we often omit the word abelian), an $M$-flow $\varphi$ on a directed graph $G=(V,E)$ is a mapping $E\to M$ such that the oriented sum around every vertex $v$ is zero:

[TABLE]

We say that $M$-flow $\varphi$ is an $(M,B)$-flow if $\varphi(e)\in B$ for all $e\in E$ (we always assume that $B\subseteq M$ and that $B$ is symmetric, i. e. $B=-B$).

An $M$-tension $\tau$ is again a mapping $E\to M$ but the condition is that the oriented sum along every cycle $C$ is zero, explicitly

[TABLE]

where $C^{+}$ are edges of $C$ with one orientation along the cycle and $C^{-}$ the edges with the opposite orientation. We define $(M,B)$-tension to be an $M$-tension which uses only values from a symmetric set $B\subseteq M$.

For planar graphs, flows and tensions are dual notions – every flow in primal graph induces a tension in its dual and vice versa. A tension can be equivalently described by a mapping $p\colon V\to M$ (usually called a group coloring or a potential). The value $\tau(uv)$ is defined as $p(v)-p(u)$, we write $\tau=\delta p$. Note that $p$ is nothing more than a homomorphism into Cayley graph of $M$. Because a composition of homomorphisms is a homomorphism, the following statement holds:

Observation 1**.**

Let $M$, $M^{\prime}$ be abelian groups and $B\subset M$, $B^{\prime}\subset M^{\prime}$ their symmetric subsets. If there is a graph homomorphism from ${\rm Cay}(M,B)$ into ${\rm Cay}(M^{\prime},B^{\prime})$, then every graph with an $(M,B)$-tension has an $(M^{\prime},B^{\prime})$-tension.

Many questions about flows on graphs were motivated by mimicking the properties of coloring in the dual setting [7]. In the same spirit, we ask for the dual version of Observation 1:

Conjecture 2** (DeVos [1]).**

Let $M$, $M^{\prime}$ be abelian groups and $B\subset M$, $B^{\prime}\subset M^{\prime}$ their symmetric subsets. If there is a graph homomorphism from ${\rm Cay}(M,B)$ into ${\rm Cay}(M^{\prime},B^{\prime})$, then every graph with an $(M,B)$-flow has an $(M^{\prime},B^{\prime})$-flow.

This is still an open problem but it holds in some special cases (the first three appear in [1], the last one is probably new).

• •

If $G$ is planar (because of duality and Observation 1).

• •

If $0\in B^{\prime}$ (every graph has an $(M^{\prime},\left\{0\right\})$-flow).

• •

If $B=M\setminus\left\{0\right\}$ and $B^{\prime}=M^{\prime}\setminus\left\{0\right\}$: Here an $(M,B)$-flow is just a nowhere-zero $M$-flow. It is known that the existence of a nowhere-zero flow is monotone in the size of the group [7].

• •

If $M={\mathbb{Z}}_{2n+1}$, $B=\left\{n,n+1\right\}$, $M^{\prime}={\mathbb{Z}}_{2n-1}$, $B^{\prime}=\left\{n-1,n\right\}$: In this case ${\rm Cay}(M,B)\cong C_{2n+1}$ has a graph homomorphism to ${\rm Cay}(M^{\prime},B^{\prime})\cong C_{2n-1}$, we will show how to transform an $(M,B)$-flow into an $(M^{\prime},B^{\prime})$. Let $f$ be an $(M,B)$-flow on a graph $G$. It is known [6] that $G$ also has an integer flow $f^{\prime}$ such that for every edge $e$ we have $|f^{\prime}(e)|<2n+1$ and $f(e)\equiv f^{\prime}(e)\pmod{2n+1}$. This means that $f^{\prime}(e)\in\{\pm n,\pm(n+1)\}$, in other words $f^{\prime}$ is a nowhere-zero fractional $\frac{2n+1}{n}$-flow. In the other direction it is easy to see that a graph with a fractional $\frac{2n+1}{n}$-flow also has an $(M,B)$-flow. It is known [2] that the existence of a nowhere-zero fractional $\frac{2n+1}{n}$-flow implies the existence of a nowhere-zero fractional $\frac{2n-1}{n-1}$-flow.

2 New Framework

The structure of homomorphism from ${\rm Cay}(M,B)$ to ${\rm Cay}(M^{\prime},B^{\prime})$ is hard to describe. Instead we take any mapping $m\colon M\to M^{\prime}$ (not necessarily a group homomorphism) and let $B^{\prime}$ be determined by $m$ (so $B^{\prime}$ is the minimal set for which $m$ is a graph homomorphism). This is achieved by the following technical definition:

Definition 3**.**

Let $M,M^{\prime}$ be abelian groups and $m\colon M\to M^{\prime}$ any mapping. For $x\in M$ we define its homomorphic image

[TABLE]

We omit the index $m$ whenever possible. Observe that in the case of tensions ${\mathcal{H}}(x)$ is exactly the set of possible images of value $x$ on some edge after composing original tension represented by a group coloring with $m$:

Observation 4**.**

Let $p\colon V\to M$ be a group coloring and let $m\colon M\to M^{\prime}$ be any mapping between abelian groups $M$ and $M^{\prime}$. Define $p^{\prime}=m\circ p$, $\tau=\delta p$, and $\tau^{\prime}=\delta p^{\prime}$. Then

• •

$\tau$* is a $M$-tension,*

• •

$\tau^{\prime}$* is a $M^{\prime}$-tension, and*

• •

$\forall e\in E:\tau^{\prime}(e)\in{\mathcal{H}}(\tau(e)).$**

Conjecture 5** (Reformulation of Conjecture 2).**

Let $M$ and $M^{\prime}$ be abelian groups and $B$ a symmetric subset of $M$. For any mapping $m\colon M\to M^{\prime}$ every graph with an $(M,B)$-flow has an $(M^{\prime},\bigcup_{x\in B}{\mathcal{H}}(x))$-flow.

The traditional approach to solving flow-related conjectures is to study properties of hypothetical minimal counterexample. Usually the problem is reduced to cubic graphs by splitting / decontracting vertices. This, however, is not possible with Conjecture 2 because decontracting a vertex may create an edge with a new value found nowhere else, modifying $B$. To overcome this we formulated the following property:

Property 6** (Strong homomorphism property).**

Let $G$ be a (directed) graph and $M$ an abelian group. We say that $G$ has strong homomorphism property (SHP) for $M$ if for every mapping $m\colon M\to M^{\prime}$ (where $M^{\prime}$ is any abelian group) and every $M$-flow $\varphi$ there exists a $M^{\prime}$-flow $\varphi^{\prime}$ such that $\varphi^{\prime}(e)\in{\mathcal{H}}(\varphi(e))$ for all edges $e$. We say that $G$ has SHP if it has SHP for all abelian groups.

Note that SHP allows the flow to be zero on some edges but such edges are not interesting because ${\mathcal{H}}(0)$ is always $\left\{0\right\}$. The SHP allows us to study only cubic graphs – we can make any graph (sub)cubic by decontracting its vertices of high degree and if SHP holds for such a decontracted graphs then it holds for the original graph too. To state this in a formal way, we need the following technical definition:

Definition 7**.**

We say that digraph a $H$ is a cubification of digraph $G$ if $H$ can be obtained from $G$ using following operations:

decontraction of vertex of degree at least 4 (such that both new vertices have degree at least 3), 2. 2.

suppression of a vertex of degree 2, 3. 3.

deletion of a bridge, 4. 4.

deletion of a loop, and 5. 5.

deletion of an isolated vertex.

With this definition we want to show that every non-cubic graph can be reduced to a smaller cubic one. To get this we need to use a slightly non-standard definition of the size of the graph which considers graphs with larger degrees bigger. Suitable definition for us is

[TABLE]

Observation 8** (Reducibility of SHP to cubic graphs).**

Let $M$ be an abelian group and let $G$ be a digraph. If some cubification of $G$ has SHP for $M$, then also $G$ has SHP for $M$. Moreover for every flow $\varphi:E\to M$ there exists a non-strictly smaller (possibly empty) cubic graph $G^{\prime}$ and a nowhere-zero flow $\varphi^{\prime}:E^{\prime}\to M$ such that if SHP does not hold for $\varphi$ on $G$ then SHP also does not hold for $\varphi^{\prime}$ on $G^{\prime}$.

Proof.

To prove the first part, we only need to show that inverse of each operation used in Definition 7 does not break SHP:

Suppose $G=G_{1}/e$ and $G_{1}$ has SHP for $M$, let $m:M\to M^{\prime}$ be any mapping. Let $\varphi$ be an $M$-flow on $G$. There is a unique extension of $\varphi$ to $G_{1}$, we use $\varphi$ for this extension as well. (Note that the value of $\varphi(e)$ may be [math].) As $G_{1}$ has SHP for $M$, there is an $M^{\prime}$-flow $\varphi^{\prime}$ on $G_{1}$ such that $\varphi^{\prime}(e)\in{\mathcal{H}}_{m}(\varphi(e))$. The restriction of $\varphi^{\prime}$ to $G=G_{1}/e$ is the desired $M^{\prime}$-flow on $G$. 2. 2.

Subdivision of an edge is obvious when the new vertex of degree 2 has both in-degree and out-degree 1. In the other case SHP still holds because ${\mathcal{H}}(-x)=-{\mathcal{H}}(x)$. 3. 3.

Addition of a bridge does not break SHP because flow on a bridge is always 0 and ${\mathcal{H}}(0)=\left\{0\right\}$. 4. 4.

Addition of a loop is also simple because ${\mathcal{H}}(x)$ is always non-empty and we can assign any value on a loop without affecting the rest of the flow. 5. 5.

Addition of an isolated vertex does not change the flow at all.

The moreover part: Note that $\Phi=\sum_{v\in V}3^{\deg v}$ for every cubification is strictly smaller than $\Phi$ of the original graph. To obtain $G^{\prime}$ we set $G^{\prime}=G$, $\varphi^{\prime}=\varphi$, and apply following operations as long as possible:

Remove an edge $e^{\prime}\in E^{\prime}$ such that $\varphi^{\prime}(e^{\prime})=0$. 2. 2.

Apply some cubification operation on $G^{\prime}$.

Because each of the operations decreases $\Phi(G^{\prime})$, the process terminates. If the resulting $\varphi^{\prime}$ was not nowhere-zero, we still could remove an edge with 0 flow, and if $G^{\prime}$ was not cubic, we could get a non-trivial cubification. ∎

The SHP is a natural strengthening of Conjecture 5 – we just fix a particular $(M,B)$-flow $\varphi$ and try to find a $(M^{\prime},B^{\prime})$-flow $\varphi^{\prime}$ with an extra requirement $\varphi^{\prime}(e)\in{\mathcal{H}}(\varphi(e))$. Observation 4 shows that a variation of SHP for tensions holds in general, so also all planar graphs have SHP due to duality.

With computer aid we found out that SHP does not hold in general. The smallest counterexample we found is $K_{3,3}$ with a particular ${\mathbb{Z}}_{5}$-flow and the universal mapping (see Figure 1 and Definition 11 below). Although SHP does not hold for $K_{3,3}$ in general it still holds for groups ${\mathbb{Z}}_{3}$ and ${\mathbb{Z}}_{4}$ because SHP always holds for groups of size at most 4 (Theorem 14). We also tested that SHP holds for Petersen graph and ${\mathbb{Z}}_{5}$ (we did not try larger snarks due to computational complexity). This motivates our next conjecture:

Conjecture 9** (SHP for minimal groups).**

For every graph $G$ the strong homomorphism property holds for group ${\mathbb{Z}}_{k}$ where $k$ is minimal such that $G$ admits a nowhere-zero ${\mathbb{Z}}_{k}$-flow.

It is easy to observe that SHP holds for $m$ which are (induced by) a group homomorphism but a more general statement is true:

Observation 10**.**

Let $G$ be a graph and let $m\colon M\to M^{\prime}$ be some mapping of abelian groups. Let $h\colon M^{\prime}\to M^{\prime\prime}$ be a group homomorphism. If SHP (resp. Conjecture 5) holds for $G$ and $m$ then it also holds for $G$ and $h\circ m$.

Proof.

Let $\varphi^{\prime}$ be an $M^{\prime}$-flow guaranteed by SHP. For a group homomorphism $h$ holds ${\mathcal{H}}_{h}(x)=\left\{h(x)\right\}$. Then $\varphi^{\prime\prime}=h\circ\varphi^{\prime}$ is also an $M^{\prime\prime}$-flow and its values satisfy

[TABLE]

3 Universal objects

Observation 10 leads us to the definition of a universal mapping such that if SHP (or Conjecture 5) holds for this mapping, it also holds for every other mapping.

Definition 11** (Universal mapping).**

Let $M$ be an abelian group. We define its universal group $\mathcal{G}_{M}={\mathbb{Z}}^{M}$ and its universal mapping $\mathcal{M}_{M}\colon M\to\mathcal{G}_{M}$:

[TABLE]

where $g_{x}$ is a vector with $1$ on position $x$ and [math] elsewhere.

The group ${\mathbb{Z}}^{M\setminus\left\{0\right\}}$ (with 0 mapped to 0 instead of $g_{0}$) would be sufficient but we choose the definition with ${\mathbb{Z}}^{M}$ to simplify the proofs. Note that with this definition $\mathcal{G}_{M}$ is just a free group generated by elements of $M$.

Observation 12**.**

The universal mapping is universal for both Conjecture 5 and SHP, i. e., if for a given graph (and flow) Conjecture 5 (resp. SHP) holds for the universal mapping then it holds for every mapping.

Proof.

Let $m\colon M\to M^{\prime}$ be any mapping. We can also interpret $m$ as a homomorphism $m_{\rm ext}\colon\mathcal{G}_{M}\to M^{\prime}$ – mapping $m$ defines values of generators $e_{x}$ and hence it can be uniquely extended into mapping on the whole group which is a homomorphism (here we are using the fact that $kg_{x}=0\Rightarrow k=0$ for $x\neq 0$). Moreover $m=\mathcal{M}_{M}\circ m_{\rm ext}$ so Observation 10 finishes the proof. ∎

There also exists a universal object on the left-hand side – a universal flow $\mathcal{F}$ (which is just the flow into a free group generated by edges outside of some fixed spanning tree) – but $\mathcal{G}_{\mathcal{F}}$ has infinitely many generators so we have not found any reasonable way to work with it. Also note that although the universal group is infinite, SHP holds for the universal group ${\mathbb{Z}}^{M}$ if and only if it holds for ${\mathbb{Z}}_{k}^{M}$ for any $k>\Delta(G)$.

4 Partial results

In this section we prove SHP for some special cases of the mapping or the group.

Theorem 13** (Mappings with one “hole”).**

Strong homomorphism property holds for mappings $m\colon{\mathbb{Z}}_{k}\to{\mathbb{Z}}_{l}$ defined by $m(x)=ax\bmod l$ where $a\in{\mathbb{Z}}$. (Here we interpret elements of ${\mathbb{Z}}_{k}$ as integers $0,1,2,\dots,k-1$.)

Proof.

Note that mapping $m$ is a composition of mappings $m_{1}\colon{\mathbb{Z}}_{k}\to{\mathbb{Z}}$ defined by $m_{1}(x)=x$ and $m_{2}\colon{\mathbb{Z}}\to{\mathbb{Z}}_{l}$ defined by $m_{2}(x)=ax\bmod l$, and that $m_{2}$ is a group homomorphism. Hence we only need to show that SHP holds for $m_{1}$, the rest follows from Observation 10.

So we need to show that for every ${\mathbb{Z}}_{k}$-flow $\varphi$ there exists a ${\mathbb{Z}}$-flow $\varphi^{\prime}$ such that $\varphi^{\prime}(e)\in{\mathcal{H}}_{m_{1}}(\varphi(e))=\left\{\varphi(e),\varphi(e)-k\right\}$. This, however, is a well-known result of Tutte [6]. ∎

Theorem 14**.**

Strong homomorphism property holds for groups ${\mathbb{Z}}_{2}$, ${\mathbb{Z}}_{3}$, ${\mathbb{Z}}_{2}^{2}$, and ${\mathbb{Z}}_{4}$.

Proof.

Due to Observation 8 we know that minimal counter-example is a cubic graph $G$ and nowhere-zero flow $\varphi$. We denote the generators of the right-hand side free group $a$, $b$, $c$, …

• •

${\mathbb{Z}}_{2}$: The only graph cubic graph with nowhere-zero ${\mathbb{Z}}_{2}$-flow is the empty graph, for which the claim holds.

• •

${\mathbb{Z}}_{3}$: Let the mapping $m$ be $0\mapsto a$, $1\mapsto b$, and $2\mapsto c$ so ${\mathcal{H}}(1)=\left\{b-a,c-b,a-c\right\}$. A cubic graph has nowhere-zero ${\mathbb{Z}}_{3}$-flow if and only if it is bipartite. So we make all edges directed from one partition to the other and split them into 3 perfect matchings. Observe that either $\varphi\equiv 1$ or $\varphi\equiv 2$ in which case we flip the orientation of edges to get the $\varphi\equiv 1$. We assign one of the following flow values to each matching: $b-a$, $c-b$, $a-c$.

• •

${\mathbb{Z}}_{2}^{2}$: Let the mapping $m$ be $00\mapsto a$, $01\mapsto b$, $10\mapsto c$, and $11\mapsto d$. Then

[TABLE]

Let $C_{1},C_{2},C_{3}\colon E\to\left\{0,\pm 1\right\}$ be a 3-CDC of $G$ defined (here we slightly abuse notation and define $e\in C\Leftrightarrow C(e)\neq 0$):

[TABLE]

And we define $\psi\colon E\to{\mathbb{Z}}^{4}$. Recall that $a=(1,0,0,0)\in{\mathbb{Z}}^{4}$, and $b$, $c$, $d$ are defined similarly.

[TABLE]

It is easy to check that $\psi$ is a ${\mathbb{Z}}^{4}$-flow and $\psi(e)\in{\mathcal{H}}(\varphi(e))$.

• •

${\mathbb{Z}}_{4}$: We observe that every vertex (with all incident edges in same direction) has either values $2,1,1$ or $2,3,3$. Hence edges with value 2 are a perfect matching. When we remove them we obtain disjoint union of circuits and we modify orientation of remaining edges so they are directed along circuits. With this orientation values around every vertex are $1,2,3$ so both edges with value 1 and edges with value 3 are a perfect matching.

Let $m$ be $0\mapsto a$, $1\mapsto b$, $2\mapsto c$, and $3\mapsto d$. Then

[TABLE]

Let $C_{1},C_{2},C_{3}\colon E\to\left\{0,\pm 1\right\}$ be a 3-CDC of $G$ defined:

[TABLE]

Observe that we can choose orientation of $C_{2}$ such that no edge has value $-1$ in $C_{2}$. And we define $\psi\colon E\to{\mathbb{Z}}^{4}$. Recall that $a=(1,0,0,0)\in{\mathbb{Z}}^{4}$, and $b$, $c$, $d$ are defined similarly.

[TABLE]

It is easy to check that $\psi$ is a ${\mathbb{Z}}^{4}$-flow and $\psi(e)\in{\mathcal{H}}(\varphi(e))$. ∎

5 Connection to CDC

Flows obtained from SHP or Conjecture 5 can be easily transformed into an oriented cycle double cover. Cycle Double Cover Conjecture (CDC, [5, 3]) is together with 3-flow, 4-flow and 5-flow conjectures one of the major open questions in the field of group flows. Moreover if $G$ has SHP for $M$ and a nowhere-zero $M$-flow then the obtained CDC is orientable and has only $\left|M\right|$ cycles. This increases importance of Conjecture 5 and of determining for which graphs and groups does SHP hold.

Theorem 15** (Universal group and CDC).**

Let $M$ be any abelian group. If a graph $G$ has a flow in $\mathcal{G}_{M}$ using only values $\bigcup_{x\in M\setminus\left\{0\right\}}{\mathcal{H}}_{\mathcal{M}_{M}}(x)$ then it has an orientable cycle double cover using $\left|M\right|$ cycles.

Proof.

Denote $H=\bigcup_{x\in M\setminus\left\{0\right\}}{\mathcal{H}}(x)$. Observe that all elements of $H$ are of form $g_{a}-g_{b}$ for some $a,b\in M$ and those $a,b$ are unique. Fix an $H$-flow $\varphi$. We define directed cycles (as mappings $E\to\left\{-1,0,1\right\}$) $\mathcal{C}_{x}(a):=(\varphi(a))_{x}$ and claim that $C=\left\{\mathcal{C}_{x}\right\}_{x\in M}$ is an orientable cycle double cover. From definition $C$ covers each edge twice, once in each direction, and every $\mathcal{C}_{x}$ is a flow with values $\left\{-1,0,1\right\}$ because it is a composition of a flow and group homomorphism so it is a cycle. ∎

Seymour in 1981 [4] proved that every graph without a bridge admits nowhere-zero ${\mathbb{Z}}_{6}$-flow which combined with the previous theorem for ${\mathbb{Z}}_{6}$ gives us the following corollary.

Corollary 16**.**

Conjecture 2 implies that every bridgeless graph has an orientable cycle double cover with at most 6 cycles.