[Paper Review] Star Edge Coloring of the Cartesian Product of Graphs

Behnaz Omoomi, Marzieh Vahid Dastjerdi|arXiv (Cornell University)|Feb 5, 2018

Advanced Graph Theory Research10 references3 citations

TL;DR

This paper establishes tight upper bounds for the star chromatic index of Cartesian products of graphs, particularly determining the exact star chromatic index of 2D grids and providing bounds for d-dimensional grids, hypercubes, and toroidal grids. It introduces a novel star compatibility framework and leverages structural properties of cycles and paths to derive results, with key findings including χ′s(Pm □ Pn) = 5 for m,n ≥ 2 and χ′s(Qd) ≤ 2d−2, which is tight for d=3,4.

ABSTRACT

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we first obtain some upper bounds for the star chromatic index of the Cartesian product of two graphs. We then determine the exact value of the star chromatic index of $2$-dimensional grids. We also obtain some upper bounds on the star chromatic index of the Cartesian product of a path with a cycle, $d$-dimensional grids, $d$-dimensional hypercubes and $d$-dimensional toroidal grids, for every positive integer $d$.

Motivation & Objective

• Address the lack of precise bounds on the star chromatic index for Cartesian products of graphs, especially grids and hypercubes.
• Establish tight upper bounds for the star chromatic index of d-dimensional grids, hypercubes, and toroidal grids.
• Provide exact values for the star chromatic index of 2D grids (Pm □ Pn) and partial results for Pm □ Cn.
• Introduce and apply the concept of star compatibility to derive new upper bounds for Cartesian products.
• Conjecture that the star chromatic index of Cm □ Cn is at most 7 for all m,n ≥ 3.

Proposed method

• Propose a new framework based on star compatibility to bound the star chromatic index of G □ H using the star chromatic indices and chromatic numbers of G and H.
• Use Theorem 1 to derive a general upper bound: χ′s(G □ H) ≤ min{χ′s(G)χ(H) + χ′s(H), χ′s(H)χ(G) + χ′s(G)}.
• Apply the concept of (k,r)-star colorability to cycles, showing even cycles are (4,2)-star colorable and odd cycles (except C3) are (7,3)-star colorable.
• Use inductive reasoning and known base cases (e.g., χ′s(C4 □ C4) = 6) to bound the star chromatic index of higher-dimensional hypercubes.
• Apply Corollary 1 and Theorem 2 to extend bounds to d-dimensional toroidal grids, distinguishing cases based on cycle parity.
• Use structural coloring patterns and case analysis to derive bounds for Cm □ Cn depending on the parity of m and n.

Experimental results

Research questions

• RQ1 What is the exact value of the star chromatic index for the 2D grid Pm □ Pn for all m,n ≥ 2?
• RQ2 How does the star chromatic index of Pm □ Cn behave, and what are its exact values for infinite families of m and n?
• RQ3 What upper bounds can be established for the star chromatic index of d-dimensional grids, hypercubes, and toroidal grids?
• RQ4 What is the relationship between the star chromatic index of Cartesian products and the star chromatic indices of their factors?
• RQ5 Can the star chromatic index of Cm □ Cn be bounded uniformly, and is it possible that χ′s(Cm □ Cn) ≤ 7 for all m,n ≥ 3?

Key findings

• The star chromatic index of the 2D grid Pm □ Pn is exactly 5 for all integers m,n ≥ 2.
• For the Cartesian product Pm □ Cn, the star chromatic index is at most 7, with exact values determined for infinite families of m and n.
• The star chromatic index of the d-dimensional hypercube Qd is at most 2d−2, and this bound is tight for d=3 and d=4.
• The star chromatic index of the Cartesian product of two even cycles Cm □ Cn is at most 7, and for odd cycles, it is at most 10.
• The d-dimensional toroidal grid Tl₁,l₂,…,ld is (4d,2)-star colorable when all li are even, with χ′s ≤ 4d−1.
• The d-dimensional toroidal grid Tl₁,l₂,…,ld is (7d,3)-star colorable when all li > 3, yielding χ′s ≤ 7d−4.