Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The IC-Indices of Complete Bipartite Graphs. | The IC-Indices of Complete Bipartite Graphs Chin-Lin Shiue Department of Applied Mathematics Chung Yuan Christian University Chung Li Taiwan 32023. email clshiue@. Hung-Lin Fuy Department of Applied Mathematics National Chiao Tung University Hsin Chu Taiwan 30050. email hlfu@. Submitted Sep 6 2007 Accepted Mar 5 2008 Published Mar 12 2008 Mathematics Subject Classifications 05C78 Abstract Let G be a connected graph and let f be a function mapping V G into N. We define f H v2V H f v for each subgraph H of G. The function f is called an IC-coloring of G if for each integer k in the set 1 2 f G g there exists an induced connected subgraph H of G such that f H k and the IC-index of G M G is the maximum value of f G where f is an IC-coloring of G. In this paper we show that M Km n 3 2m n 2 2m 2 2 for each complete bipartite graph Km n 2 m n. 1 Introduction Given a connected graph G. Let f be a function mapping V G into N. We define f H Xv2V H f v for each subgraph H of G. Then f is called an IC-coloring of G if for each integer k in the set 1 f G 1 2 f G g there exists an induced connected subgraph H of G such that f H k. Clearly the constant function f v 1 for each v 2 V G is an IC-coloring in which f G V G . It is interesting to know the maximum value of f G such that f is an IC-coloring of G. This maximum value is defined as the IC-index of G denoted by M G . We say that f is a maximal IC-coloring of G if f is an IC-coloring of G with f G M G . Research supported by NSC 95-2115-M-033-005. yResearch supported by NSC 95-2115-M-009-006. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R43 1 The study of the IC-index of a graph originated from the so-called postage stamp problem in Number Theory which has been extensively studied in the literature 1 6v9 11 13v16 . In 1992 G. Chappel formulated IC-colorings as subgraph sums problem and he observed the IC-index of cycle Cn is bounded above by n2 n 1 . M Cn n2 n 1. Later in 1995 Penrice 12
