TY - JOUR

T1 - A stability theorem on fractional covering of triangles by edges

AU - Haxell, Penny

AU - Kostochka, Alexandr

AU - Thomassé, Stéphan

N1 - Funding Information:
The first author’s research was partially supported by NSERC . The research of the second author was supported in part by NSF grants DMS-06-50784 and DMS-0965587 and by grant 08-01-00673 from the Russian Foundation for Basic Research .

PY - 2012/7

Y1 - 2012/7

N2 - Let ν(G) denote the maximum number of edge-disjoint triangles in a graph G and τ *(G) denote the minimum total weight of a fractional covering of its triangles by edges. Krivelevich proved that τ *(G)≤2ν(G) for every graph G. This is sharp, since for the complete graph K 4 we have ν(K 4)=1 and τ *(K 4)=2. We refine this result by showing that if a graph G has τ *(G)ge2ν(G)-x, then G contains ν(G)-⌊10x⌋ edge-disjoint K 4-subgraphs plus an additional ⌊10x⌋ edge-disjoint triangles. Note that just these K 4's and triangles witness that τ *(G)ge2ν(G)-⌊10x⌋ Our proof also yields that τ *(G)≤1.8ν(G) for each K 4-free graph G. In contrast, we show that for each ε>0, there exists a K 4-free graph G ε such that τ(G ε)>(2-ε)ν(G ε).

AB - Let ν(G) denote the maximum number of edge-disjoint triangles in a graph G and τ *(G) denote the minimum total weight of a fractional covering of its triangles by edges. Krivelevich proved that τ *(G)≤2ν(G) for every graph G. This is sharp, since for the complete graph K 4 we have ν(K 4)=1 and τ *(K 4)=2. We refine this result by showing that if a graph G has τ *(G)ge2ν(G)-x, then G contains ν(G)-⌊10x⌋ edge-disjoint K 4-subgraphs plus an additional ⌊10x⌋ edge-disjoint triangles. Note that just these K 4's and triangles witness that τ *(G)ge2ν(G)-⌊10x⌋ Our proof also yields that τ *(G)≤1.8ν(G) for each K 4-free graph G. In contrast, we show that for each ε>0, there exists a K 4-free graph G ε such that τ(G ε)>(2-ε)ν(G ε).

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U2 - 10.1016/j.ejc.2011.09.024

DO - 10.1016/j.ejc.2011.09.024

M3 - Article

AN - SCOPUS:84856973198

SN - 0195-6698

VL - 33

SP - 799

EP - 806

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 5

ER -