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A polynomial time algorithm to find star chromatic index on bounded treewidth graphs with given maximum degree

Yichen Wang, Mei Lu

TL;DR

For a bounded treewidth graph with given maximum degree, it is shown that the star edge coloring problem can be solved in polynomial time.

Abstract

A star edge coloring of a graph $G$ is a proper edge coloring with no 2-colored path or cycle of length four. The star edge coloring problem is to find an edge coloring of a given graph $G$ with minimum number $k$ of colors such that $G$ admits a star edge coloring with $k$ colors. This problem is known to be NP-complete. In this paper, for a bounded treewidth graph with given maximum degree, we show that it can be solved in polynomial time.

A polynomial time algorithm to find star chromatic index on bounded treewidth graphs with given maximum degree

TL;DR

For a bounded treewidth graph with given maximum degree, it is shown that the star edge coloring problem can be solved in polynomial time.

Abstract

A star edge coloring of a graph is a proper edge coloring with no 2-colored path or cycle of length four. The star edge coloring problem is to find an edge coloring of a given graph with minimum number of colors such that admits a star edge coloring with colors. This problem is known to be NP-complete. In this paper, for a bounded treewidth graph with given maximum degree, we show that it can be solved in polynomial time.
Paper Structure (4 sections, 4 theorems, 9 equations, 1 figure)

This paper contains 4 sections, 4 theorems, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. The star edge coloring algorithm
  4. Discussion

Key Result

Theorem 3.1

For everg graph $G$ of order $n$ with treewidth $k$ and maximum degree $\Delta$, and integer $c$, there is a deterministic algorithm that determines in time $O(nc^{2{(k+1)}^2\Delta^6})$ whether $G$ has a star edge coloring using at most $c$ colors or not and finds such star edge coloring if it exist

Figures (1)

  • Figure 1: Five cases of an invalid but proper star coloring in which the bold vertices belong to $X_i - \{v'\}$. The two endpoints may be the same and then it results in a bicolored $C_4$ rather than $P_4$. Blue and red represent two different colors and $L$ (resp. $R$) represents that the edge is from $G_L'$ (resp. $G_R'$). The bold vertices must be in $X_i$ from previous facts.

Theorems & Definitions (6)

  • Definition 2.1
  • Theorem 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Conjecture 4.1