On Validity of Reed Conjecture for Classes of Graphs with Two Forbidden Subgraphs

TL;DR

This paper proves Reed Conjecture holds for specific classes of graphs characterized by two forbidden subgraphs, advancing understanding of the conjecture's validity in restricted graph families.

Contribution

The paper establishes the validity of Reed Conjecture for four new classes of graphs defined by two forbidden subgraphs, expanding the known cases where the conjecture holds.

Findings

Reed Conjecture is valid for {P4UnionK1, Kite}-free graphs.

Reed Conjecture is valid for {Chair, Kite}-free graphs.

Reed Conjecture is valid for {K2UnionK2complement, H}-free graphs and {2K2, M}-free graphs.

Abstract

Reed Conjecture is open for more than 20 years now. Here we prove that Reed Conjecture is valid for (1) {P4UnionK1, Kite}-free graphs (2) {Chair, Kite}-free graphs (3) {K2UnionK2complement , H}-free graphs and (4) {2K2, M}-free graphs where H and M are graphs on six vertices each. Reed conjecture is still open in general.