[Paper Review] List strong edge coloring of some classes of graphs

Watcharintorn Ruksasakchai, Tao Wang|arXiv (Cornell University)|Feb 23, 2014

Graph Labeling and Dimension Problems3 citations

TL;DR

This paper establishes tight upper bounds for the list strong chromatic index of graphs with bounded maximum degree and structural constraints. It proves that for graphs with maximum degree Δ ≤ 4 and maximum average degree < 3, the list strong chromatic index is at most 3Δ + 1; for planar graphs with Δ ≥ 4 and girth ≥ 7, it is at most 3Δ, improving prior bounds via discharging arguments and structural analysis.

ABSTRACT

A {\em strong edge coloring} of a graph is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} of a graph is the minimum number of colors needed to obtain a strong edge coloring. In an analogous way, we can define the list version of strong edge coloring and list version of strong chromatic index. In this paper, we prove that if $G$ is a graph with maximum degree at most four and maximum average degree less than $3$, then the list strong chromatic index is at most $3Δ+ 1$, where $Δ$ is the maximum degree of $G$. In addition, we prove that if $G$ is a planar graph with maximum degree at least $4$ and girth at least $7$, then the list strong chromatic index is at most $3Δ$.

Motivation & Objective

• To determine tight upper bounds for the list strong chromatic index in sparse and planar graph classes.
• To extend known results on strong edge coloring to the list coloring setting, where edge colors are chosen from predefined lists.
• To improve existing upper bounds for planar graphs with high girth and bounded degree using discharging methods.
• To investigate the structural constraints that allow for tighter bounds on list strong edge coloring.
• To resolve open questions on the tightness of upper bounds in the list strong edge coloring setting.

Proposed method

• Employing a discharging method to analyze vertex and face charges in planar graphs with girth ≥ 7 and maximum degree Δ ≥ 4.
• Defining and applying a set of discharging rules (R1–R10) to redistribute initial charges and prove that all vertices and faces end with non-negative final charge.
• Using structural claims (Claims 1–8) to restrict possible configurations of low-degree vertices and their neighbors, enabling charge redistribution.
• Applying the discharging technique to show that the minimum number of colors required in a list strong edge coloring is bounded by 3Δ or 3Δ+1.
• Leveraging known results on subcubic graphs and planar graph properties (e.g., Grötzsch’s theorem) as foundational tools.
• Combining extremal graph theory with list coloring techniques to derive bounds that are tight under given constraints.

Experimental results

Research questions

• RQ1 What is the list strong chromatic index of graphs with maximum degree at most four and maximum average degree less than three?
• RQ2 Can the list strong chromatic index of planar graphs with girth at least seven and maximum degree at least four be bounded by 3Δ?
• RQ3 How do structural constraints such as girth and maximum average degree affect the list strong edge coloring number?
• RQ4 Are the upper bounds of 3Δ+1 and 3Δ for the list strong chromatic index tight, or can they be further improved?
• RQ5 What role do discharging rules and vertex/face charge redistribution play in proving tight bounds for list strong edge coloring?

Key findings

• For graphs with maximum degree Δ ≤ 4 and maximum average degree < 3, the list strong chromatic index is at most 3Δ + 1.
• For planar graphs with maximum degree Δ ≥ 4 and girth at least 7, the list strong chromatic index is at most 3Δ.
• The discharging method successfully proves that all vertices and faces in such graphs end with non-negative final charge, confirming the bounds.
• The results improve upon previous bounds, such as 3Δ + 5 and 3Δ + 3, for planar graphs with girth ≥ 6 and ≥ 7, respectively.
• The paper provides the first tight upper bounds for list strong edge coloring in these graph classes, suggesting potential for further improvement.
• No example was found to demonstrate the tightness of the bounds, indicating that the upper bounds may be improvable.