# Homomorphisms

## Cores of Cayley graphs ★★

Author(s): Samal

\begin{conjecture} Let $M$ be an abelian group. Is the \Def[core]{core (graph theory)} of a \Def{Cayley graph} (on some power of $M$) a Cayley graph (on some power of $M$)? \end{conjecture}

Keywords: Cayley graph; core

## Pentagon problem ★★★

Author(s): Nesetril

\begin{question} Let $G$ be a 3-regular graph that contains no cycle of length shorter than $g$. Is it true that for large enough~$g$ there is a \Def[homomorphism]{graph_homomorphism} $G \to C_5$? \end{question}

Keywords: cubic; homomorphism

## Mapping planar graphs to odd cycles ★★★

Author(s): Jaeger

\begin{conjecture} Every planar graph of girth $\ge 4k$ has a homomorphism to $C_{2k+1}$. \end{conjecture}

Keywords: girth; homomorphism; planar graph

## Weak pentagon problem ★★

Author(s): Samal

\begin{conjecture} If $G$ is a cubic graph not containing a triangle, then it is possible to color the edges of $G$ by five colors, so that the complement of every color class is a bipartite graph. \end{conjecture}

## Algorithm for graph homomorphisms ★★

Author(s): Fomin; Heggernes; Kratsch

\begin{question} % Enter your conjecture in LaTeX % You may change "conjecture" to "question" or "problem" if you prefer. Is there an algorithm that decides, for input graphs $G$ and $H$, whether there exists a homomorphism from $G$ to $H$ in time $O(c^{|V(G)|+|V(H)|})$ for some constant $c$? \end{question}

## Circular choosability of planar graphs ★

Author(s): Mohar

Let $G = (V, E)$ be a graph. If $p$ and $q$ are two integers, a $(p,q)$-colouring of $G$ is a function $c$ from $V$ to $\{0,\dots,p-1\}$ such that $q \le |c(u)-c(v)| \le p-q$ for each edge $uv\in E$. Given a list assignment $L$ of $G$, i.e.~a mapping that assigns to every vertex $v$ a set of non-negative integers, an $L$-colouring of $G$ is a mapping $c : V \to N$ such that $c(v)\in L(v)$ for every $v\in V$. A list assignment $L$ is a $t$-$(p,q)$-list-assignment if $L(v) \subseteq \{0,\dots,p-1\}$ and $|L(v)| \ge tq$ for each vertex $v \in V$ . Given such a list assignment $L$, the graph G is $(p,q)$-$L$-colourable if there exists a $(p,q)$-$L$-colouring $c$, i.e. $c$ is both a $(p,q)$-colouring and an $L$-colouring. For any real number $t \ge 1$, the graph $G$ is $t$-$(p,q)$-choosable if it is $(p,q)$-$L$-colourable for every $t$-$(p,q)$-list-assignment $L$. Last, $G$ is circularly $t$-choosable if it is $t$-$(p,q)$-choosable for any $p$, $q$. The circular choosability (or circular list chromatic number or circular choice number) of G is $$cch(G) := \inf\{t \ge 1 : G \text{ is circularly t-choosable}\}.$$

\begin{problem} What is the best upper bound on circular choosability for planar graphs? \end{problem}

Keywords: choosability; circular colouring; planar graphs