[Paper Review] A survey on star edge-coloring of graphs

Hui Lei, Yongtang Shi|arXiv (Cornell University)|Sep 17, 2020

Graph Labeling and Dimension Problems37 references4 citations

TL;DR

This survey provides a comprehensive overview of star edge-coloring in graphs, defining the star chromatic index as the minimum number of colors needed to color edges such that no path or cycle of length four is bicolored. It synthesizes key results on subcubic, bipartite, planar, and product graphs, presents tight bounds for complete and grid graphs, and identifies open problems including the true order of magnitude of the star chromatic index for complete graphs.

ABSTRACT

The star chromatic index of a multigraph $G$, denoted $χ'_{st}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bicolored. We survey the results of determining the star chromatic index, present the interesting proofs and techniques, and collect many open problems and conjectures.

Motivation & Objective

• To systematize and summarize known results on the star chromatic index across various graph classes.
• To present and analyze key proofs and techniques used in establishing upper and lower bounds for star edge-coloring.
• To collect and highlight open problems and conjectures, particularly regarding the asymptotic growth of the star chromatic index for complete graphs.
• To explore algorithmic complexity and list coloring variants of star edge-coloring.
• To extend results to non-standard graph families such as Halin graphs, generalized Petersen graphs, and graph products.

Proposed method

• Defining star edge-coloring as a proper edge-coloring where no path or cycle of length four is bicolored.
• Using the equivalence between star edge-coloring of a graph G and star vertex-coloring of its line graph L(G).
• Applying extremal combinatorics and arithmetic progression-free sets to derive upper bounds for complete graphs.
• Employing double counting techniques to establish lower bounds on the star chromatic index.
• Reducing known NP-complete problems (e.g., 3-edge-coloring of cubic graphs) to prove NP-completeness of determining star chromatic index ≤3.
• Analyzing structural properties of specific graph families (e.g., grids, coronas, graph joins) to derive exact or tight bounds on the star chromatic index.

Experimental results

Research questions

• RQ1 What is the true asymptotic order of magnitude of the star chromatic index for complete graphs K_n?
• RQ2 Is χ′_st(K_n) = O(n), or does it grow faster, such as n^{1+o(1)}?
• RQ3 What is the computational complexity of determining whether a graph admits a star edge-coloring with at most k colors?
• RQ4 How does the star chromatic index behave for special graph families like subcubic, bipartite, planar, and product graphs?
• RQ5 Can the list version of star edge-coloring be characterized or bounded for general graphs?

Key findings

• For complete graphs K_n, the star chromatic index satisfies 2n(1+o(1)) ≤ χ′_st(K_n) ≤ n · 2^{2√2(1+o(1))√log n} / (log n)^{1/4}, with the upper bound derived using sets without arithmetic progressions.
• For any graph G with maximum degree Δ, χ′_st(G) ≤ Δ · 2^{O(1)√log Δ}, representing the best known general upper bound.
• It is NP-complete to determine whether χ′_st(G) ≤ 3 for an arbitrary graph G, even when Δ(G) = 3.
• For the Cartesian product of cycles C_m □ C_n, χ′_st(C_m □ C_n) is either 6 or 7, with exact conditions provided; it is conjectured that for large m,n, the index is 6.
• For the corona product P_m ◦ P_n, χ′_st(P_m ◦ P_n) = n+2 for m ≥ 3, and similar exact values are established for other corona products.
• For graph joins P_n ◦ P_2, χ′_st(P_n ◦ P_2) = (3n-1)/2 + 4 when n ≥ 3 and odd, and (3n)/2 + 3 when n ≥ 4 and even, with χ′_st(P_3 ◦ P_3) = 10.