At Least Half Of All Graphs Satisfy \chi \leq {1/4}\omega + {3/4}\Delta + 1

TL;DR

This paper proves that for any graph, either the graph or its complement satisfies a specific chromatic number bound involving clique number and maximum degree, with implications for self-complementary graphs.

Contribution

It establishes a new chromatic bound that applies to all graphs or their complements, including self-complementary graphs, advancing understanding of graph coloring properties.

Findings

At least one of a graph or its complement satisfies the chromatic bound.

Self-complementary graphs meet the established chromatic bound.

The bound relates chromatic number, clique number, and maximum degree.

Abstract

We prove that for any graph G at least one of G or $\bar{G}$ satisfies $\chi \leq {1/4}\omega + {3/4}\Delta + 1$. In particular, self-complementary graphs satisfy this bound.