Potentially H-bigraphic sequences

• Volume: 29, Issue: 3, page 583-596
• ISSN: 2083-5892

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Abstract

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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 $\sigma \left({K}_{s,t},m,n\right)$, σ(Pₜ,m,n) and $\sigma \left({C}_{2t},m,n\right)$.

How to cite

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Michael Ferrara, et al. "Potentially H-bigraphic sequences." Discussiones Mathematicae Graph Theory 29.3 (2009): 583-596. <http://eudml.org/doc/270910>.

@article{MichaelFerrara2009,
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 $σ(K_\{s,t\},m,n)$, σ(Pₜ,m,n) and $σ(C_\{2t\},m,n)$.},
author = {Michael Ferrara, Michael Jacobson, John Schmitt, Mark Siggers},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {degree sequence; bipartite graph; potential number},
language = {eng},
number = {3},
pages = {583-596},
title = {Potentially H-bigraphic sequences},
url = {http://eudml.org/doc/270910},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Michael Ferrara
AU - Michael Jacobson
AU - John Schmitt
AU - Mark Siggers
TI - Potentially H-bigraphic sequences
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 3
SP - 583
EP - 596
AB - 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 $σ(K_{s,t},m,n)$, σ(Pₜ,m,n) and $σ(C_{2t},m,n)$.
LA - eng
KW - degree sequence; bipartite graph; potential number
UR - http://eudml.org/doc/270910
ER -

References

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7. [7] J. Li, Z. Song and R. Luo, The Erdös-Jacobson-Lehel conjecture on potentially Pₖ-graphic sequences is true, Science in China (A) 41 (1998) 510-520, doi: 10.1007/BF02879940. Zbl0906.05031
8. [8] J. Li and J. Yin, The smallest degree sum that yields potentially K_{r,r}-graphic sequences, Science in China (A) 45 (2002) 694-705. Zbl1099.05505
9. [9] J. Li and J. Yin, An extremal problem on potentially K_{r,s}-graphic sequences, Discrete Math. 260 (2003) 295-305, doi: 10.1016/S0012-365X(02)00765-3. Zbl1017.05055
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12. [12] K. Zarankiewicz, Problem P 101, Colloq. Math. 2 (1951) 301.

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