Some notes on the signed bad number in bipartite graphs

TL;DR

This paper investigates the signed bad number and negative decision number in bipartite graphs, correcting previous bounds and providing more general bounds for triangle-free graphs using Mantel's theorem.

Contribution

It corrects inaccurate bounds from prior work and introduces more general bounds for triangle-free graphs, characterizing those attaining these bounds.

Findings

Previous bounds for bipartite graphs are invalid.

New bounds are established for triangle-free graphs.

Characterization of graphs attaining the bounds is provided.

Abstract

In this paper, we deal with the signed bad number and the negative decision number of graphs. We show that two upper bounds concerning these two parameters for bipartite graphs in papers [Discrete Math. Algorithms Appl. 1 (2011), 33--41] and [Australas. J. Combin. 41 (2008), 263--272] are not true as they stand. We correct them by presenting more general bounds for triangle-free graphs by using the classic theorem of Mantel from the extremal graph theory and characterize all triangle-free graphs attaining these bounds.