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Obstructions for homomorphisms to odd cycles in series-parallel graphs

Eun-Kyung Cho, Ilkyoo Choi, Boram Park, Mark Siggers

Abstract

For a graph $H$, an $H$-colouring of a graph $G$ is a vertex map $φ:V(G) \to V(H)$ such that adjacent vertices are mapped to adjacent vertices. A graph $G$ is $C_{2k+1}$-critical if $G$ has no $C_{2k+1}$-colouring but every proper subgraph of $G$ has a $C_{2k+1}$-colouring. We prove a structural characterisation of $C_{2k+1}$-critical graphs when $k \geq 2$. In the case that $k = 2$, we use the aforementioned charazterisation to show a $C_3$-free series-parallel graph $G$ has a $C_5$-colouring if either $G$ has neither $C_8$ nor $C_{10}$, or $G$ has no two $5$-cycles sharing a vertex.

Obstructions for homomorphisms to odd cycles in series-parallel graphs

Abstract

For a graph , an -colouring of a graph is a vertex map such that adjacent vertices are mapped to adjacent vertices. A graph is -critical if has no -colouring but every proper subgraph of has a -colouring. We prove a structural characterisation of -critical graphs when . In the case that , we use the aforementioned charazterisation to show a -free series-parallel graph has a -colouring if either has neither nor , or has no two -cycles sharing a vertex.

Paper Structure

This paper contains 6 sections, 10 theorems, 18 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Obstructions to $C_5$-colouring
  4. Proof of Theorem \ref{['lem:families:ccF']}
  5. Proof of \ref{['thm:main']} and \ref{['cor:twoC5s']}
  6. Generalisation to $(2k+1)$-cycles for $k \geq 2$

Key Result

Lemma 2.2

For a positive integer $k$ and a $C_{2k+1}$-critical series-parallel graph $G$, the following hold:

Figures (3)

  • Figure 1: Base graphs -- graphs in $\cup \mathscr{F}$ with at most 10 vertices.
  • Figure 2: Illustrations for $H_2 \mathbin{\raisebox{0.105ex}{origin=c]{180}{$\uplus$}}} (H_4+H_5)$ and $H_5 \mathbin{\raisebox{0.105ex}{origin=c]{180}{$\uplus$}}} (H_4 + H_5)$.
  • Figure 3: Illustrations for $H_2 \mathbin{\raisebox{0.105ex}{origin=c]{180}{$\uplus$}}} H_5$ and $H_3 \mathbin{\raisebox{0.105ex}{origin=c]{180}{$\uplus$}}} H_5$

Theorems & Definitions (28)

  • Conjecture 1.1
  • Example 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 3.5
  • proof
  • ...and 18 more