$C_{2k+1}$-coloring of bounded-diameter graphs

Abstract

For a fixed graph $H$, in the graph homomorphism problem, denoted by $Hom(H)$, we are given a graph $G$ and we have to determine whether there exists an edge-preserving mapping $\varphi: V(G) \to V(H)$. Note that $Hom(C_3)$, where $C_3$ is the cycle of length $3$, is equivalent to $3$-Coloring. The question whether $3$-Coloring is polynomial-time solvable on diameter-$2$ graphs is a well-known open problem. In this paper we study the $Hom(C_{2k+1})$ problem on bounded-diameter graphs for $k\geq 2$, so we consider all other odd cycles than $C_3$. We prove that for $k\geq 2$, the $Hom(C_{2k+1})$ problem is polynomial-time solvable on diameter-$(k+1)$ graphs -- note that such a result for $k=1$ would be precisely a polynomial-time algorithm for $3$-Coloring of diameter-$2$ graphs. Furthermore, we give subexponential-time algorithms for diameter-$(k+2)$ graphs. We complement these results with a lower bound for diameter-$(2k+2)$ graphs -- in this class of graphs the $Hom(C_{2k+1})$ problem is NP-hard and cannot be solved in subexponential-time, unless the ETH fails. Finally, we consider another direction of generalizing $3$-Coloring on diameter-$2$ graphs. We consider other target graphs $H$ than odd cycles but we restrict ourselves to diameter $2$. We show that if $H$ is triangle-free, then $Hom(H)$ is polynomial-time solvable on diameter-$2$ graphs.

Figures (3)

• Figure 1: Case $k=4$. Orange vertices denote a list of some vertex $v\in V_3$, blue vertices denote all possible lists of a neighbor $u$ of $v$ when \ref{['red:edges']} cannot be applied, i.e., every vertex of $L(u)$ is a neighbor of a vertex of $L(v)$ and every vertex of $L(v)$ is a neighbor of a vertex of $L(u)$.
• Figure 2: The copy of $H'$ in $G$ and a vertex $v$ such that for some homomorphism $\varphi$, it holds $\varphi(v_i)=\varphi(v)$. We show that a vertex $u$ which is a common neighbor of $v$ and some neighbor $v_j$ of $v_i$ in the copy of $H'$ cannot exist.
• Figure 3: Construction of $G$ for $k=2$ and clause $\gamma_j=(\neg x_1\lor x_2 \lor \neg x_3)$. Green vertices belong to the cycle $C$, blue vertices are those introduced for variables, and yellow ones are those introduced for the clause $\gamma_j$. Remaining vertices belong to paths $P_{ij}$.

Theorems & Definitions (36)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5: Edwards DBLP:journals/tcs/Edwards86
• Lemma 1: DBLP:journals/combinatorica/FederHH99
• Lemma 2
• proof
• Lemma 3
• proof