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

For a coloring c of a connected graph G, let Pi = (C-1, C-2,..., C-k) be an ordered partition of V(G) into the resulting color classes. For a vertex upsilon of G, the color code c(Pi)(upsilon) of v is the ordered k-tuple (d(upsilon, C-1), d(upsilon, C-2),..., d(upsilon, C-k)), where d(upsilon, C-i) = min{d(upsilon, x): x is an element of C-i} for 1 less than or equal to i less than or equal to k. If distinct vertices have distinct color codes, then c is called a locating-coloring. The locating-chromatic number chi(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 greater than or equal to 3 containing an induced complete multipartite subgraph of order n - 1, then (n + 1)/2 less than or equal to chi(L)(G) n and, furthermore, for each integer k with (n + 1)/2 less than or equal to k less than or equal to n, there exists such a graph G of order n with chi(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 greater than or equal to 4 with locating-chromatic number n - 1. (C) 2003 Elsevier B.V. All rights reserved.