Tuza's Conjecture for Threshold Graphs

TL;DR

This paper proves Tuza's conjecture for threshold graphs and certain co-chain graphs, expanding the classes of graphs where the conjecture is confirmed, and providing new insights into graph triangle deletion problems.

Contribution

The paper establishes Tuza's conjecture for threshold graphs and specific co-chain graphs, extending known cases and contributing to the understanding of triangle removal in dense graphs.

Findings

Tuza's conjecture holds for threshold graphs.

Confirmed the conjecture for co-chain graphs with equal-sized parts divisible by 4.

Extended the classes of graphs where the conjecture is verified.

Abstract

Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including planar graphs. However, for dense graphs that are neither cliques nor 4-colorable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4.