An Optimal χ-Bound for (P_6, diamond)-Free Graphs

09/03/2018
by   Kathie Cameron, et al.
0

Given two graphs H_1 and H_2, a graph G is (H_1,H_2)-free if it contains no induced subgraph isomorphic to H_1 or H_2. Let P_t be the path on t vertices and K_t be the complete graph on t vertices. The diamond is the graph obtained from K_4 by removing an edge. In this paper we show that every (P_6, diamond)-free graph G satisfies χ(G)<ω(G)+3, where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. Our bound is attained by the complement of the famous 27-vertex Schläfli graph. Our result unifies previously known results on the existence of linear χ-binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect (P_6, diamond)-free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well-known characterization of 3-colourable (P_6,K_3)-free graphs.

READ FULL TEXT
research
06/16/2021

Colouring graphs with no induced six-vertex path or diamond

The diamond is the graph obtained by removing an edge from the complete ...
research
09/27/2017

Linearly χ-Bounding (P_6,C_4)-Free Graphs

Given two graphs H_1 and H_2, a graph G is (H_1,H_2)-free if it contains...
research
04/06/2021

Coloring graph classes with no induced fork via perfect divisibility

For a graph G, χ(G) will denote its chromatic number, and ω(G) its cliqu...
research
04/03/2020

Wheel-free graphs with no induced five-vertex path

A 4-wheel is the graph consisting of a chordless cycle on four vertices ...
research
12/28/2018

Towards a constructive formalization of Perfect Graph Theorems

Interaction between clique number ω(G) and chromatic number χ(G) of a ...
research
05/16/2022

Optimal chromatic bound for (P_2+P_3, P̅_̅2̅+̅ ̅P̅_̅3̅)-free graphs

For a graph G, let χ(G) (ω(G)) denote its chromatic (clique) number. A P...
research
11/25/2021

Variants of the Gyàrfàs-Sumner Conjecture: Oriented Trees and Rainbow Paths

Given a finite family ℱ of graphs, we say that a graph G is "ℱ-free" if ...

Please sign up or login with your details

Forgot password? Click here to reset