On the locating chromatic number of Kneser graphs

TL;DR

This paper investigates the locating chromatic number of Kneser graphs, establishing exact values for certain cases and bounds for others, advancing understanding of graph coloring properties.

Contribution

It provides exact locating chromatic numbers for Kneser graphs with k=2 and bounds for general n and k, extending previous graph coloring research.

Findings

chi_L(KG(n,2))=n-1 for all n.

chi_L(KG(n,k)) n-1 when n k^2.

Bounds for the locating chromatic number of odd graphs are presented.

Abstract

Let $c$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(C_1,C_2,...,C_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $\Pi$ is defined to be the ordered $k$-tuple $$c_{{}_\Pi}(v):=(d(v,C_1),d(v,C_2),...,d(v,C_k)),$$ where $d(v,C_i)=\min\{d(v,x) |x\in C_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $c$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by $\Cchi_{{}_L}(G)$. In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results we show that $\Cchi_{{}_L}(KG(n,2))=n-1$ for all $n\geq 5$. Then, we prove that $\Cchi_{{}_L}(KG(n,k))\leq n-1$, when $n\geq k^2$. Moreover, we present some bounds for the locating chromatic number of odd graphs.