Discussiones Mathematicae Graph Theory 29(3) (2009) 583-596
doi: 10.7151/dmgt.1466

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POTENTIALLY H-BIGRAPHIC SEQUENCES

Michael Ferrara

University of Colorado at Denver
e-mail: michael.ferrara@ucdenver.edu

Michael Jacobson

University of Colorado at Denver
e-mail: michael.jacobson@ucdenver.edu

John Schmitt

Middlebury College
e-mail: jschmitt@middlebury.edu

Mark Siggers

Kyungpook National University
e-mail: mhsiggers@knu.ac.kr

Abstract

We extend the notion of a potentially H-graphic sequence as follows. Let A and B be nonnegative integer sequences. The sequence pair S = (A,B) is said to be bigraphic if there is some bipartite graph G = (X∪Y,E) such that A and B are the degrees of the vertices in X and Y, respectively. If S is a bigraphic pair, let σ(S) denote the sum of the terms in A.

Given a bigraphic pair S, and a fixed bipartite graph H, we say that S is potentially H-bigraphic if there is some realization of S containing H as a subgraph. We define σ(H,m,n) to be the minimum integer k such that every bigraphic pair S = (A,B) with |A|= m, |B|= n and σ(S) ≥k is potentially H-bigraphic. In this paper, we determine σ(Ks,t,m,n), σ(Pt,m,n) and σ(C2t,m,n).

Keywords: degree sequence, bipartite graph, potential number.

2000 Mathematics Subject Classification: 05C07, 05C35.

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Received 5 June 2008
Accepted 5 September 2008>