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Sudoku Number of Corona of Graphs

Manju S Nair, Aparna Lakshmanan S, S Arumugam

TL;DR

The paper studies Sudoku coloring in graphs by formalizing the Sudoku number $sn(G)$ as the minimum order of an induced subgraph $G[S]$ whose $k$-coloring extends uniquely to a $k$-coloring of $G$, using the notion of uniquely color extendable (u.c.e) vertices. It develops bounds for corona products $G \circ H$, showing $nm \le sn(G \circ H) \le nm + sn(G)$ when $\chi(G) \ge 3$ and $|V(H)|=m \le \chi(G)-2$, with tightness via explicit constructions. The paper also computes exact $sn(\cdot)$ values for several corona families, including $C_n \circ K_1$, $W_n \circ K_1$, $K_n \circ K_m$, and $C_n \circ P_m$, and discusses the graphs attaining the bounds. These results deepen understanding of Sudoku-type colorings in graph products and connect to related list-coloring phenomena.

Abstract

Let $G = (V,E)$ be a graph of order $n$ with chromatic number $χ(G) = k$, let $S \subset V$ and let $C_0$ be a $k$-coloring of the induced subgraph $G[S]$. The coloring $C_0$ is called an extendable coloring, if $C_0$ can be extended to a $k$-coloring of $G$ and it is a Sudoku coloring of $G$ if the extension is unique. The smallest order of such an induced subgraph $G[S]$ of $G$ which admits a Sudoku coloring is called the Sudoku number of $G$ and is denoted by $sn(G)$. In this paper, we first introduce the notion of uniquely color extendable vertex and then we obtain the lower and upper bounds for the Sudoku number of $G \circ K_1$. Some families of graphs which attain these bounds are also obtained. The exact value of the Sudoku number of corona of $C_n$, $W_n$ and $K_n$ with $K_1$ and $C_n \circ P_m$ are also obtained.

Sudoku Number of Corona of Graphs

TL;DR

The paper studies Sudoku coloring in graphs by formalizing the Sudoku number as the minimum order of an induced subgraph whose -coloring extends uniquely to a -coloring of , using the notion of uniquely color extendable (u.c.e) vertices. It develops bounds for corona products , showing when and , with tightness via explicit constructions. The paper also computes exact values for several corona families, including , , , and , and discusses the graphs attaining the bounds. These results deepen understanding of Sudoku-type colorings in graph products and connect to related list-coloring phenomena.

Abstract

Let be a graph of order with chromatic number , let and let be a -coloring of the induced subgraph . The coloring is called an extendable coloring, if can be extended to a -coloring of and it is a Sudoku coloring of if the extension is unique. The smallest order of such an induced subgraph of which admits a Sudoku coloring is called the Sudoku number of and is denoted by . In this paper, we first introduce the notion of uniquely color extendable vertex and then we obtain the lower and upper bounds for the Sudoku number of . Some families of graphs which attain these bounds are also obtained. The exact value of the Sudoku number of corona of , and with and are also obtained.
Paper Structure (2 sections, 10 theorems, 14 equations, 12 figures)

This paper contains 2 sections, 10 theorems, 14 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Corona Product of Graphs

Key Result

Lemma 1.1

Let $G$ be a graph with $\chi(G) \geq 3$. Suppose $C_0$ is an extendable coloring of $G[S]$ for $S \subset V(G)$. If there is a pendant vertex $v \notin S$, then $C_0$ is not a Sudoku coloring maria2023sudoku.

Figures (12)

  • Figure 1: Line graph of $C_n \circ K_1$
  • Figure 2: Uniquely extendable coloring of corona of $L(C_3 \circ K_1)$ and $K_1$
  • Figure 3: Uniquely extendable coloring of corona of $L(C_4 \circ K_1)$ and $K_1$
  • Figure 4: Uniquely extendable coloring of $C_5 \circ K_1$ and its final coloring
  • Figure 5: Uniquely extendable coloring of $W_4 \circ K_1$
  • ...and 7 more figures

Theorems & Definitions (18)

  • Definition 1.1
  • Lemma 1.1
  • Definition 1.2
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 8 more