On a conjecture about the strong odd chromatic number of planar graphs

TL;DR

This work investigates the strong odd chromatic number $\chi_{so}(G)$ of planar graphs, driven by Pang's conjecture that $\chi_{so}$ could be bounded by 13. The authors analyze joins of cycles with independent sets and generalized one-point unions to obtain exact values and structural formulas, notably showing $\chi_{so}(I_y(G_m,G_n)) = \chi_{so}(W_m) + \chi_{so}(W_n) - 1$ and deriving wheel-graph bounds. Using these results, they construct infinite families of planar graphs with $\chi_{so}$ ranging from 14 up to 17, including $\chi_{so}(I_y(G_8,G_8)) = 17$, thereby providing counterexamples to the 13-bound conjecture. The findings demonstrate that planar graphs can have substantially higher strong odd chromatic numbers than previously believed, and they open avenues for tighter bounds and further exploration of graph-construction operations in this coloring paradigm.

Abstract

A proper coloring of a graph $G$ is said to be a strong odd coloring of $G$, if for every vertex $v$ and every color $c$, either $c$ appears on an odd number of vertices in the neighborhood of $v$ or $c$ is absent in the neighborhood of $v$. The strong odd chromatic number of $G$ is defined as the smallest integer $k$ for which $G$ admits a strong odd coloring using $k$ colors. In this paper, we evaluate the strong odd chromatic number of join of cycles and empty graphs and one point union of graphs. Using these results, we construct infinite family of planar graphs that serves as counter examples to a recent conjecture regarding the upper bound of the strong odd chromatic number of planar graphs.

Figures (2)

• Figure 1: Construction of $H$ with $\chi_{\text{so}}(H) = \chi(C_4)$
• Figure 2: Planar graph $G$ with $\chi_{\text{so}}(G) = 17$

Theorems & Definitions (8)

• Definition
• Theorem 1
• proof
• Theorem 2
• Definition
• Definition
• Theorem 3
• proof