The profile of the Cartesian product of graphs

Authors:
David Kuo;Jing-Ho Yan
Affiliations:
Department of Applied Mathematics, Dong Hwa University, Hualien 974, Taiwan;Department of Mathematics, Aletheia University, Tamsui 251, Taiwan
Venue:
Discrete Applied Mathematics
Year:
2008

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Abstract

Given a graph G, a proper labelingf of G is a one-to-one function from V(G) onto {1,2,...,|V(G)|}. For a proper labeling f of G, the profile widthw"f(v) of a vertex v is the minimum value of f(v)-f(x), where x belongs to the closed neighborhood of v. The profile of a proper labelingfofG, denoted by P"f(G), is the sum of all the w"f(v), where v@?V(G). The profile ofG is the minimum value of P"f(G), where f runs over all proper labeling of G. In this paper, we show that if the vertices of a graph G can be ordered to satisfy a special neighborhood property, then so can the graph GxQ"n. This can be used to determine the profile of Q"n and K"mxQ"n.