Maximum odd induced subgraph of a graph concerning its chromatic number

• Published in: Journal of graph theory Vol. 107; no. 3; pp. 578 - 596
• Main Authors: Wang, Tao, Wu, Baoyindureng
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.11.2024
• Subjects: Apexes; chromatic number; Graph theory; Graphs; induced subgraphs; odd subgraphs
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.23148

Let f o ( G ) ${f}_{o}(G)$ be the maximum order of an odd induced subgraph of G $G$. In 1992, Scott proposed a conjecture that f o ( G ) ≥ n χ ( G ) ${f}_{o}(G)\ge \frac{n}{\chi (G)}$ for a graph G $G$ of order n $n$ without isolated vertices, where χ ( G ) $\chi (G)$ is the chromatic number of G $G$. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that f o ( G ) ≥ 2 n 4 ${f}_{o}(G)\ge 2\unicode{x0230A}\frac{n}{4}\unicode{x0230B}$ for a connected graph G $G$ of order n $n$. Scott's conjecture is open for graphs with chromatic number at least 3.