(2,1)-Total labelling of trees with sparse vertices of maximum degree

Authors:
Jing Huang;Haina Sun;Weifan Wang;Dong Chen
Affiliations:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;Department of Fundamental Courses, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China;Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Venue:
Information Processing Letters
Year:
2009

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Abstract

The (2,1)-total labelling number @l"2^t(G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices, or two adjacent edges, have the same label and the difference between the labels of a vertex and its incident edges is at least 2. Let T be a tree with maximum degree @D=4. Let D"@D(T) denote the set of integers k for which there exist two distinct vertices of maximum degree of distance at k in T. It was known that @D+1==3, there exist infinitely many trees T with @D=4 and k@?D"@D(T) such that @l"2^t(T)=@D+2.