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The palette index of the Cartesian product of paths, cycles and regular graphs

Aleksander Vesel

TL;DR

This work investigates the palette index $\check s(G)$ for Cartesian products $G \Box H$ when one factor is a path or cycle and the other factor is regular or class 1 nearly regular. It develops a range of exact values and tight bounds by leveraging edge-coloring constructions, class 1 versus class 2 dichotomies, and structural decompositions such as even cycle decompositions and partial-cube embeddings. Key contributions include exact results for $\check s(C_s \Box C_t)$ and $\check s(C_s \Box P_t)$, a general bound $\check s(G \Box H) \le \check s(G)+2$ when one factor is a cycle or path and the other regular, and the identification of conditions under which the palette index collapses to 1 or 2 for broad families (notably when one factor is class 1). The findings advance understanding of how the palette index behaves under Cartesian products and yield practical edge-coloring constructions with minimal palette diversity, with implications for combinatorial design and related applications.

Abstract

The palette of a vertex v in a graph G is the set of colors assigned to the edges incident to v. The palette index of G is the minimum number of distinct palettes among the vertices, taken over all proper edge colorings of G. This paper presents results on the palette index of the Cartesian product $G \Box H$, where one of the factor graphs is a path or a cycle. Additionally, it provides exact results and bounds on the palette index of the Cartesian product of two graphs, where one factor graph is isomorphic to a regular or class 1 nearly regular graph.

The palette index of the Cartesian product of paths, cycles and regular graphs

TL;DR

This work investigates the palette index for Cartesian products when one factor is a path or cycle and the other factor is regular or class 1 nearly regular. It develops a range of exact values and tight bounds by leveraging edge-coloring constructions, class 1 versus class 2 dichotomies, and structural decompositions such as even cycle decompositions and partial-cube embeddings. Key contributions include exact results for and , a general bound when one factor is a cycle or path and the other regular, and the identification of conditions under which the palette index collapses to 1 or 2 for broad families (notably when one factor is class 1). The findings advance understanding of how the palette index behaves under Cartesian products and yield practical edge-coloring constructions with minimal palette diversity, with implications for combinatorial design and related applications.

Abstract

The palette of a vertex v in a graph G is the set of colors assigned to the edges incident to v. The palette index of G is the minimum number of distinct palettes among the vertices, taken over all proper edge colorings of G. This paper presents results on the palette index of the Cartesian product , where one of the factor graphs is a path or a cycle. Additionally, it provides exact results and bounds on the palette index of the Cartesian product of two graphs, where one factor graph is isomorphic to a regular or class 1 nearly regular graph.
Paper Structure (5 sections, 20 theorems, 15 equations, 5 figures)

This paper contains 5 sections, 20 theorems, 15 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Palette index of Cartesian products of regular and class 1 nearly regular graphs
  4. Palette index of Cartesian products with a path and cycle
  5. Paths, cycles and regular graphs

Key Result

Proposition 2.2

If $M$ is a perfect matching of a nontrivial graph $G$, then $\check s(G) \le \check s(G - M)$.

Figures (5)

  • Figure 1: An edge coloring of a 4-regular graph $G$ with a perfect matching $M$
  • Figure 2: An edge coloring of a subgraph of $G \Box C_3$ with 2 palettes, where $G$ is a NRG derived from $Q_3$
  • Figure 3: Edge colorings of: (a) $C_5 \Box P_4$ with two palettes, (b) $C_4 \Box P_5$ with two palettes (c) $C_5 \Box C_5$ with three palettes.
  • Figure 4: An even cycle decomposition of $C_{13} \Box C_{5}$
  • Figure 5: An (incomplete) edge coloring of the Cartesian product of the Petersen graph and triangle with three palettes

Theorems & Definitions (29)

  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.3
  • Theorem 3.4
  • ...and 19 more