Graphs of order n with locating-chromatic number n−1

Abstract

For a coloring c of a connected graph G, let Π=( 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 c Π ( v) of v is the ordered k-tuple (d(v,C 1),d(v,C 2),…,d(v,C k)), where d(v,C i)= min{d(v,x) : x∈C i} for 1⩽ i⩽ k. If distinct vertices have distinct color codes, then c is called a locating-coloring. The locating-chromatic number χ L ( G) is the minimum number of colors in a locating-coloring of G. It is shown that if G is a connected graph of order n⩾3 containing an induced complete multipartite subgraph of order n−1, then ( n+1)/2⩽ χ L ( G)⩽ n and, furthermore, for each integer k with ( n+1)/2⩽ k⩽ n, there exists such a graph G of order n with χ L ( G)= k. Graphs of order n containing an induced complete multipartite subgraph of order n−1 are used to characterize all connected graphs of order n⩾4 with locating-chromatic number n−1.