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Packing Graphs of Bounded Codegree

Part of: Graph theory

Published online by Cambridge University Press:  22 March 2018

WOUTER CAMES VAN BATENBURG  and
ROSS J. KANG
WOUTER CAMES VAN BATENBURG
Affiliation:
Department of Mathematics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, Netherlands (e-mail: w.camesvanbatenburg@math.ru.nl, ross.kang@gmail.com)
ROSS J. KANG
Affiliation:
Department of Mathematics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, Netherlands (e-mail: w.camesvanbatenburg@math.ru.nl, ross.kang@gmail.com)
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Abstract

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Two graphs G1 and G2 on n vertices are said to pack if there exist injective mappings of their vertex sets into [n] such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that if (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1, then G1 and G2 pack. We consider the validity of this assertion under the additional assumption that G1 or G2 has bounded codegree. In particular, we prove for all t ⩾ 2 that if G1 contains no copy of the complete bipartite graph K2,t and Δ(G1) > 17t · Δ(G2), then (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1 implies that G1 and G2 pack. We also provide a mild improvement if moreover G2 contains no copy of the complete tripartite graph K1,1,s, s ⩾ 1.

MSC classification

Primary: 05C70: Factorization, matching, partitioning, covering and packing
Secondary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 5 , September 2018 , pp. 725 - 740
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2018

Footnotes

†

Supported by NWO grant 613.001.217.

‡

Supported by a NWO Vidi grant (639.032.614).

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