The star edge coloring of cubic Halin graphs with star chromatic index $5$

TL;DR

This work analyzes star edge coloring of cubic Halin graphs, proving a universal lower bound $\chi'_{st}(G) \ge 5$ and establishing exact values $\chi'_{st}(G)=5$ for two broad families: complete cubic Halin graphs (excluding the necklace $N_{e2}$) and Halin graphs whose characteristic trees are caterpillars (excluding $N_{e2}$). The authors develop inductive, constructive colorings that extend from smaller base graphs to larger structures, using explicit color assignments to ensure no bi-chromatic $4$-paths or cycles arise. These results provide tight bounds for large, natural families and suggest that many cubic Halin graphs have star chromatic index 5, while leaving open a full characterization of all such graphs. The techniques combine structural decompositions of the characteristic trees with careful edge-coloring constructions to achieve the bound $5$.

Abstract

The star chromatic index of a graph $G$, denoted by $χ'_{st}(G) $, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. Casselgren et al. and Hou et al. independently proved that the star chromatic index of a cubic Halin graph, except a special graph, is at most $6$. It remains an open problem to determine which of such graphs have star chromatic index $5$. In this paper, we show that if $G\ne N_{e_2}$ is a cubic Halin graph whose tree is a caterpillar or a complete tree, then $χ'_{st}(G)=5$.

Figures (10)

• Figure 1: a-b
• Figure 2: Structure of $G^{i+1}$.
• Figure 3: $G_l,l=1,2,3$
• Figure 4: a-e
• Figure 5: a-b

Theorems & Definitions (7)

• Conjecture 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 3.1
• Theorem 3.2
• Theorem 4.1
• Corollary 4.2