The k-Sudoku Number of Graphs

TL;DR

This paper introduces the concept of the $k$- Sudoku number of a graph, focusing on its properties for bipartite graphs and establishing conditions for specific values, thereby advancing understanding of unique graph colorings.

Contribution

It defines the $k$- Sudoku number, computes it for various bipartite graphs, and characterizes when this number attains certain values, also exploring its relation to supergraphs.

Findings

Computed the 3-Sudoku number for specific bipartite graphs.

Established conditions for bipartite graphs to have $sn(G,3)$ equal to $n$, $n-1$, or $n-2$.

Analyzed the relationship between the $k$- Sudoku number of a graph and its supergraphs.

Abstract

Let $G=(V,E)$ be a graph of order $n$ with chromatic number $\chi(G)$. Let $ k \geq \chi(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 $k=\chi(G)$, 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 $G$.