Decompositions of complete graphs and complete bipartite graphs into isomorphic supersubdivision graphs

Abstract

A graph H is called a supersubdivison of a graph G if H is obtained from G by replacing every edge uv of G by a complete bipartite graph K 2, m ( m may vary for each edge) by identifying u and v with the two vertices in K 2, m that form one of the two partite sets. We denote the set of all such supersubdivision graphs by SS( G) . Then, we prove the following results. 1. Each non-trivial connected graph G and each supersubdivision graph H∈ SS( G) admits an α-valuation. Consequently, due to the results of Rosa (in: Theory of Graphs, International Symposium, Rome, July 1966, Gordon and Breach, New York, Dunod, Paris, 1967, p. 349) and El-Zanati and Vanden Eynden (J. Combin. Designs 4 (1996) 51), it follows that complete graphs K 2 cq+1 and complete bipartite graphs K mq, nq can be decomposed into edge disjoined copies of H∈ SS( G) , for all positive integers m, n and c, where q=| E( H)|. 2. Each connected graph G and each supersubdivision graph in SS( G) is strongly n-elegant, where n=| V( G)| and felicitous. 3. Each supersubdivision graph in EASS( G) , the set of all even arbitrary supersubdivision graphs of any graph G, is cordial. Further, we discuss a related open problem.