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The k-Sudoku Number of Graphs

Manju S Nair, Aparna Lakshmanan S, S Arumugam

TL;DR

This work investigates the $k$-Sudoku number, focusing on $k=3$ for bipartite graphs, to understand when a small induced subgraph coloring uniquely extends to a full $3$-coloring. It provides exact $3$-Sudoku numbers for standard bipartite families such as $P_n$, $C_{2n}$, $K_{m,n}$, and $B_{m,n}$, and analyzes how the parameter behaves under corona operations and edge-attached cliques. The authors supply necessary and sufficient conditions for $sn(G,3)$ to equal $n$, $n-1$, or $n-2$, and explore relations between $sn(G,k)$ and Sudoku numbers of supergraphs, offering general bounds and constructive techniques. These results advance understanding of Sudoku-coloring extendability in graphs and inform how graph operations influence the Sudoku-number landscape.

Abstract

Let $G=(V,E)$ be a graph of order $n$ with chromatic number $χ(G)$. Let $ k \geq χ(G) $ and $S \subseteq V$. 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 $k$- Sudoku coloring of $G$, if $C_0$ can be uniquely extended to a $k$-coloring of $G$. The smallest order of such an induced subgraph $G[S]$ of $G$ which admits a $k$- Sudoku coloring is called $k$- Sudoku number of $G$ and is denoted by $sn(G,k)$. When $k=χ(G)$, we call $k$- Sudoku number of $G$ as Sudoku number of $G$ and is denoted by $sn(G)$. In this paper, we have obtained the $3$- Sudoku number of some bipartite graphs $P_n$, $C_{2n}$, $K_{m,n}$, $B_{m,n}$ and $G \circ lK_1$, where $G$ is a bipartite graph and $l\geq1$. Also, we have obtained the necessary and sufficient conditions for a bipartite graph $G$ to have $sn(G,3)$ equal to $n$, $n-1$ or $n-2$. Also, we study the relation between $k$- Sudoku number of a graph $G$ and the Sudoku number of a supergraph $H$ of $G$.

The k-Sudoku Number of Graphs

TL;DR

This work investigates the -Sudoku number, focusing on for bipartite graphs, to understand when a small induced subgraph coloring uniquely extends to a full -coloring. It provides exact -Sudoku numbers for standard bipartite families such as , , , and , and analyzes how the parameter behaves under corona operations and edge-attached cliques. The authors supply necessary and sufficient conditions for to equal , , or , and explore relations between and Sudoku numbers of supergraphs, offering general bounds and constructive techniques. These results advance understanding of Sudoku-coloring extendability in graphs and inform how graph operations influence the Sudoku-number landscape.

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 can be uniquely extended to a -coloring of . The smallest order of such an induced subgraph of which admits a - Sudoku coloring is called - Sudoku number of and is denoted by . When , we call - Sudoku number of as Sudoku number of and is denoted by . In this paper, we have obtained the - Sudoku number of some bipartite graphs , , , and , where is a bipartite graph and . Also, we have obtained the necessary and sufficient conditions for a bipartite graph to have equal to , or . Also, we study the relation between - Sudoku number of a graph and the Sudoku number of a supergraph of .
Paper Structure (3 sections, 14 theorems, 2 equations, 10 figures)

This paper contains 3 sections, 14 theorems, 2 equations, 10 figures.

Table of Contents

  1. Introduction
  2. $3$- Sudoku number of some bipartite graphs
  3. Some other results

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 (10)

  • Figure 1: $3$- Sudoku coloring of $P_6$ and its final coloring
  • Figure 2: $3$- Sudoku coloring of $C_6$ and its final coloring
  • Figure 3: $3$- Sudoku coloring of $K_{1,6}$ and its final coloring
  • Figure 4: $3$- Sudoku coloring of $K_{3,4}$ and its final coloring
  • Figure 5: $3$- Sudoku coloring of $B_{3,2}$ and its final coloring
  • ...and 5 more figures

Theorems & Definitions (25)

  • Definition 1.1
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • ...and 15 more