Extremal Graphs for a Graph Packing Theorem of Sauer and Spencer

Authors:
HEMANSHU KAUL;ALEXANDR KOSTOCHKA
Affiliations:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (e-mail: hkaul@math.uiuc.edu, kostochk@math.uiuc.edu);-
Venue:
Combinatorics, Probability and Computing
Year:
2007

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Abstract

Let G1 and G2 be graphs oforder n with maximum degree Δ1 andΔ2, respectively. G1 andG2 are said to pack if there existinjective mappings of the vertex sets into [n], such thatthe images of the edge sets do not intersect. Sauer and Spencershowed that if Δ1Δ2 n/2, then G1 and G2pack. We extend this result by showing that ifΔ1Δ2 n/2, thenG1 and G2 do not pack if andonly if one of G1 or G2 is aperfect matching and the other either isKn/2·n/2 with n/2 oddor contains Kn/2 + 1.