Balanced decomposition of a vertex-colored graph

  • Authors:

  • Shinya Fujita;Tomoki Nakamigawa

  • Affiliations:

  • Department of Mathematics, Gunma National College of Technology, 580 Toriba, Maebashi 371-8530, Japan;Department of Information Science, Shonan Institute of Technology, 1-1-25 Tsujido-Nishikaigan, Fujisawa 251-8511, Japan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2008

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Abstract

A balanced vertex-coloring of a graph G is a function c from V(G) to {-1,0,1} such that @?{c(v):v@?V(G)}=0. A subset U of V(G) is called a balanced set if U induces a connected subgraph and @?{c(v):v@?U}=0. A decomposition V(G)=V"1@?...@?V"r is called a balanced decomposition if V"i is a balanced set for 1@?i@?r. In this paper, the balanced decomposition number f(G) of G is introduced; f(G) is the smallest integer s such that for any balanced vertex-coloring c of G, there exists a balanced decomposition V(G)=V"1@?...@?V"r with |V"i|@?s for 1@?i@?r. Balanced decomposition numbers of some basic families of graphs such as complete graphs, trees, complete bipartite graphs, cycles, 2-connected graphs are studied.