Note: On hamiltonian colorings for some graphs

  • Authors:

  • Yufa Shen;Wenjie He;Xue Li;Donghong He;Xiaojing Yang

  • Affiliations:

  • Department of Mathematics and Physics, Hebei Normal University of Science and Technology, Qinhuangdao 066004, PR China and Center for Mathematics of Hebei Province, Hebei Normal University, Shijia ...;Applied Mathematics Institute, Hebei University of Technology, Tianjin 300130, PR China;Applied Mathematics Institute, Hebei University of Technology, Tianjin 300130, PR China;Applied Mathematics Institute, Hebei University of Technology, Tianjin 300130, PR China;Department of Mathematics and Physics, Hebei Normal University of Science and Technology, Qinhuangdao 066004, PR China

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

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Abstract

For a connected graph G and any two vertices u and v in G, let D(u,v) denote the length of a longest u-v path in G. A hamiltonian coloring of a connected graph G of order n is an assignment c of colors (positive integers) to the vertices of G such that |c(u)-c(v)|+D(u,v)=n-1 for every two distinct vertices u and v in G. The valuehc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic numberhc(G) of G is min{hc(c)} taken over all hamiltonian colorings c of G. In this paper we discuss the hamiltonian chromatic number of graphs G with max{D(u,v)|u,v@?V(G),uv}@?n2. As examples, we determine the hamiltonian chromatic number for a class of caterpillars, and double stars.