Greedy F-colorings of graphs

Abstract

Let G=( V, E) be a connected graph. For a symmetric, integer-valued function δ on V× V, where K is an integer constant, N 0 is the set of nonnegative integers, and Z is the set of integers, we define a C-mapping F : V×V×N 0→ Z by F( u, v, m)= δ( u, v)+ m− K. A coloring c of G is an F-coloring if F( u, v,| c( u)− c( v)|)⩾0 for every two distinct vertices u and v of G. The maximum color assigned by c to a vertex of G is the value of c, and the F-chromatic number F( G) is the minimum value among all F-colorings of G. For an ordering s : v 1,v 2,…,v n of the vertices of G, a greedy F-coloring c of s is defined by (1) c( v 1)=1 and (2) for each i with 1⩽ i< n, c( v i+1 ) is the smallest positive integer p such that F( v j , v i+1 ,| c( v j )− p|)⩾0, for each j with 1⩽ j⩽ i. The greedy F-chromatic number gF( s) of s is the maximum color assigned by c to a vertex of G. The greedy F-chromatic number of G is gF( G)=min{ gF( s)} over all orderings s of V. The Grundy F-chromatic number is GF( G)=max{ gF( s)} over all orderings s of V. It is shown that gF( G)= F( G) for every graph G and every F-coloring defined on G. The parameters gF( G) and GF( G) are studied and compared for a special case of the C-mapping F on a connected graph G, where δ( u, v) is the distance between u and v and K=1+ diam G .