Validity of Borodin and Kostochka Conjecture for classes of graphs without a single, forbidden subgraph on 5 vertices

TL;DR

This paper proves the Borodin and Kostochka Conjecture for specific classes of graphs that lack certain small forbidden subgraphs, advancing understanding of graph coloring bounds.

Contribution

It establishes the conjecture's validity for (P4 union K1)-free, P5-free, chair-free graphs, and graphs with dense neighborhoods, extending prior results.

Findings

Proves the conjecture for (P4 union K1)-free graphs

Proves the conjecture for P5-free graphs

Proves the conjecture for chair-free graphs and dense neighborhood graphs

Abstract

Problem of finding an optimal upper bound for the chromatic no. of a graph is still open and very hard. Borodin and Kostochka Conjecture is still open and if proved will improve Brook bound on Chromatic no. of a graph. Here we prove Borodin & Kostochka Conjecture for (1) (P4 Union K1)-free (2) P5-free (3) Chair-free graphs and 4) graphs with dense neighbourhoods. Certain known results follow as Corollaries.