Skolem labelled graphs

Abstract

AbstractA Skolem labelled graph is a triple (G, L, d), where G = (V, E) is a graph and L:V → d, d + 1,…, d+m satisfying: 1.(a) There are exactly two vertices in V, such that L(v)=d+i, 0 ⩽i⩽m.2.(b) The distance in G between any two vertices with the same label is the value of the label.3.(c) If G' is a proper spanning subgraph of G, (G', L, d) is not a Skolem labelled graph.Note that this definition is different from the Skolem-graceful labelling of Lee, Quach and Wang (1988). When d=1 is not specified it is assumed. We shall establish the following: 1.(1) Any tree can be embedded in a Skolem labelled tree with O(v) vertices.2.(2) Any graph can be embedded as an induced subgraph in a Skolem labelled graph on O(v3) vertices.3.(3) For d=1, we exhibit a Skolem or the minimum hooked Skolem (with as few unabelled vertices as possible) labelling for paths and cycles.4.(4) For d=1 we exhibit the minimum Skolem labelled gaph containing a path or a cycle of length n as induced subgraph