A note on coloring (even-hole,cap)-free graphs
Shenwei Huang, Murilo V. G. da Silva

TL;DR
This paper proves a chromatic bound for (even-hole, cap)-free graphs using a decomposition theorem and provides a polynomial-time approximation algorithm for their coloring.
Contribution
It establishes a new chromatic bound for (even-hole, cap)-free graphs and introduces a polynomial-time 3/2-approximation coloring algorithm based on this bound.
Findings
Chromatic number bound: χ(G) ≤ ⌊3/2 ω(G)⌋ for (even-hole, cap)-free graphs.
The bound is tight, achieved by odd holes and the Hajos graph.
A polynomial-time 3/2-approximation algorithm for coloring these graphs.
Abstract
A {\em hole} is a chordless cycle of length at least four. A hole is {\em even} (resp. {\em odd}) if it contains an even (resp. odd) number of vertices. A \emph{cap} is a graph induced by a hole with an additional vertex that is adjacent to exactly two adjacent vertices on the hole. In this note, we use a decomposition theorem by Conforti et al. (1999) to show that if a graph does not contain any even hole or cap as an induced subgraph, then , where and are the chromatic number and the clique number of , respectively. This bound is attained by odd holes and the Hajos graph. The proof leads to a polynomial-time -approximation algorithm for coloring (even-hole,cap)-free graphs.
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Limits and Structures in Graph Theory
