Some Combinatorial Problems on Halin Graphs

TL;DR

This paper studies various combinatorial problems on Halin graphs, including recognition, coloring, and bounds, and provides polynomial-time algorithms for these problems.

Contribution

It introduces polynomial-time algorithms for recognizing Halin graphs and optimally coloring them, advancing understanding of their combinatorial properties.

Findings

Polynomial-time recognition algorithm for Halin graphs

Optimal coloring algorithm for Halin graphs

Chromatic bounds established for Halin graphs

Abstract

Let $T$ be a tree with no degree 2 vertices and $L(T)=\{l_1,\ldots,l_r\}, r \geq 2$ denote the set of leaves in $T$. An Halin graph $G$ is a graph obtained from $T$ such that $V(G)=V(T)$ and $E(G)=E(T) \cup \{\{l_i,l_{i+1}\} ~|~ 1 \leq i \leq r-1\} \cup \{l_1,l_r\}$. In this paper, we investigate combinatorial problems such as, testing whether a given graph is Halin or not, chromatic bounds, an algorithm to color Halin graphs with the minimum number of colors. Further, we present polynomial-time algorithms for testing and coloring problems.