Published online by Cambridge University Press: 22 March 2018
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.
Supported by NWO grant 613.001.217.
Supported by a NWO Vidi grant (639.032.614).
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