Elsevier

Discrete Mathematics

Volume 339, Issue 11, 6 November 2016, Pages 2785-2792
Discrete Mathematics

On a packing problem of Alon and Yuster

https://doi.org/10.1016/j.disc.2016.05.023Get rights and content
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Abstract

Two graphs G1 and G2, each on n vertices, pack if there exists a bijection f from V(G1) onto V(G2) such that uvE(G1) only if f(u)f(v)E(G2). In 2014, Alon and Yuster proved that, for sufficiently large n, if |E(G1)|<nδ(G2) and Δ(G2)n/200, then G1 and G2 pack. In this paper, we characterize the pairs of graphs for which the theorem of Alon and Yuster is sharp. We also prove the stronger result that for sufficiently large n, if |E(G1)|n, Δ(G2)n/60, and Δ(G1)+δ(G2)n1, then G1 and G2 pack whenever there is a vertex v1V(G1) such that d(v1)=Δ(G1) and α(G1N[v1])δ(G2).

Keywords

Graph packing
Maximum degree
Edge sum

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