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On the $δ$-chromatic numbers of the Cartesian products of graphs

Wipawee Tangjai, Witsarut Pho-on, Panupong Vichitkunakorn

TL;DR

The paper investigates the $\delta$-chromatic number $\chi_\delta(G)$, defined as the chromatic number of the $\delta$-complement $G_\delta$, for Cartesian products of graphs. It provides a structural characterization of $ (G\square H)_\delta$ in terms of $G_\delta\square H_\delta$ plus an auxiliary set $S$, and extends this to $k$-fold products, yielding criteria for when $(G_1\square\cdots\square G_k)_\delta=(G_1)_\delta\square\cdots\square(G_k)_\delta$. The authors derive sharp general bounds for $\chi_\delta$ of Cartesian products, and they give explicit, exact values for several natural graph classes, including $\chi_\delta(S_{1,m}\square S_{1,n})=mn$, $\chi_\delta(S_{1,m}\square P_n)=m\left\lceil \frac{n-2}{2}\right\rceil$, and $\chi_\delta(P_n\square P_k)=\left\lceil \frac{(n-2)(k-2)}{2}\right\rceil$ in certain ranges. These results combine structural insights with constructive colorings and clique analyses to produce tight bounds and exact values, advancing the theory of delta-chromatic numbers in product graphs.

Abstract

In this work, we study the $δ$-chromatic number of a graph which is the chromatic number of the $δ$-complement of a graph. We give a structure of the $δ$-complements and sharp bounds on the $δ$-chromatic numbers of the Cartesian products of graphs. Furthermore, we compute the $δ$-chromatic numbers of various classes of Cartesian product graphs, including the Cartesian products between cycles, paths, and stars.

On the $δ$-chromatic numbers of the Cartesian products of graphs

TL;DR

The paper investigates the -chromatic number , defined as the chromatic number of the -complement , for Cartesian products of graphs. It provides a structural characterization of in terms of plus an auxiliary set , and extends this to -fold products, yielding criteria for when . The authors derive sharp general bounds for of Cartesian products, and they give explicit, exact values for several natural graph classes, including , , and in certain ranges. These results combine structural insights with constructive colorings and clique analyses to produce tight bounds and exact values, advancing the theory of delta-chromatic numbers in product graphs.

Abstract

In this work, we study the -chromatic number of a graph which is the chromatic number of the -complement of a graph. We give a structure of the -complements and sharp bounds on the -chromatic numbers of the Cartesian products of graphs. Furthermore, we compute the -chromatic numbers of various classes of Cartesian product graphs, including the Cartesian products between cycles, paths, and stars.
Paper Structure (6 sections, 16 theorems, 15 equations, 3 figures, 1 table)

This paper contains 6 sections, 16 theorems, 15 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Structure of the $\delta$-complements of Cartesian products
  4. Bounds on the $\delta$-chromatic numbers of Cartesian products
  5. The $\delta$-chromatic numbers of the Cartesian products of some graphs
  6. Conclusion

Key Result

Theorem 3

Let $G$ and $H$ be graphs. We have $\chi(G\square H)=\max\{\chi(G),\chi(H)\}$.

Figures (3)

  • Figure 1: A proper $mn$-coloring of $(S_{1,m}\square S_{1,n})_\delta$. The vertices of the same degree in $S_{1,m}\square S_{1,n}$ are indicated by the same color and are pairwise adjacent in $(S_{1,m}\square S_{1,n})_\delta$. Each double line denotes the edges connecting the corresponding vertices between two copies of $S_{1,n}$. Note that the blue and the green will have the same degree when $m=n$.
  • Figure 2: A proper $km$-coloring of $(S_{1,m}\square P_n)_\delta$ where $k = \lceil\frac{n-2}{2}\rceil$. The vertices of the same degree in $S_{1,m}\square P_n$ are indicated by the same color and are pairwise adjacent except for the pairs with a dashed line.
  • Figure 3: A proper 10-coloring of $(P_6\square P_7)_\delta$. The vertices in $V_2$, $V_3$ and $V_4$ are shown in red, blue and black, respectively. The vertices in each $V_i$ for $i=2,3,4$ are pairwise adjacent in $(P_6\square P_7)_\delta$ except for the pairs with a dashed line.

Theorems & Definitions (31)

  • Definition 1: math10081203
  • Definition 2: Vichitkunakorn2023
  • Theorem 3: sabidussi_1957
  • Theorem 4: Vichitkunakorn2023
  • Theorem 5
  • proof
  • Corollary 6
  • Theorem 7
  • proof
  • Theorem 8
  • ...and 21 more