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Maximum odd induced subgraph of a graph concerning its chromatic number

Tao Wang, Baoyindureng Wu

Abstract

Let $f_{o}(G)$ be the maximum order of an odd induced subgraph of $G$. In 1992, Scott proposed a conjecture that $f_{o}(G)\geq \frac {n} {2χ(G)}$ for a graph $G$ of order $n$ without isolated vertices, where $χ(G)$ is the chromatic number of $G$. 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 $f_{o}(G)\geq 2\lfloor\frac {n} {4}\rfloor$ for a connected graph $G$ of order $n$. Scott's conjecture is open for a graph with chromatic number at least 3.

Maximum odd induced subgraph of a graph concerning its chromatic number

Abstract

Let be the maximum order of an odd induced subgraph of . In 1992, Scott proposed a conjecture that for a graph of order without isolated vertices, where is the chromatic number of . 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 for a connected graph of order . Scott's conjecture is open for a graph with chromatic number at least 3.
Paper Structure (5 sections, 8 theorems, 36 equations)

This paper contains 5 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Counterexamples on bipartite graphs
  3. Line graphs
  4. Disproof of a related conjecture
  5. Concluding remarks

Key Result

Lemma 2.2

Let $H$ be the Heawood graph and $G=H+M$ as shown in Fig. 2, where $M$ is the extra perfect matching depicted in red. The following statements hold: (1) $G$ is a vertex-transitive and edge-transitive 4-regular graph. (2) $G[N(x)\cup N(y)]$ is isomorphic to the graph shown in Fig. 3 for an edge $xy\i

Theorems & Definitions (18)

  • Conjecture 1.1: Caro19942
  • Conjecture 2.1: Scott Scott1992
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Theorem 3.1
  • proof
  • proof
  • ...and 8 more