TY - JOUR
T1 - Large subgraphs without short cycles
AU - Foucaud, F.
AU - Krivelevich, M.
AU - Perarnau, G.
N1 - Publisher Copyright:
© 2015 Society for Industrial and Applied Mathematics.
PY - 2015
Y1 - 2015
N2 - We study two extremal problems about subgraphs excluding a family F of graphs: (i) Among all graphs with m edges, what is the smallest size f(m, F) of a largest F-free subgraph? (ii) Among all graphs with minimum degree δ and maximum degree Δ, what is the smallest minimum degree h(δ, Δ, F) of a spanning F-free subgraph with largest minimum degree? These questions are easy to answer for families not containing any bipartite graph. We study the case where F is composed of all even cycles of length at most 2r, r ≥ 2. In this case, we give bounds on f(m, F) and h(δ, Δ, F) that are essentially asymptotically tight up to a logarithmic factor. In particular for every graph G, we show the existence of subgraphs with arbitrarily high girth and with either many edges or large minimum degree. These subgraphs are created using probabilistic embeddings of a graph into extremal graphs.
AB - We study two extremal problems about subgraphs excluding a family F of graphs: (i) Among all graphs with m edges, what is the smallest size f(m, F) of a largest F-free subgraph? (ii) Among all graphs with minimum degree δ and maximum degree Δ, what is the smallest minimum degree h(δ, Δ, F) of a spanning F-free subgraph with largest minimum degree? These questions are easy to answer for families not containing any bipartite graph. We study the case where F is composed of all even cycles of length at most 2r, r ≥ 2. In this case, we give bounds on f(m, F) and h(δ, Δ, F) that are essentially asymptotically tight up to a logarithmic factor. In particular for every graph G, we show the existence of subgraphs with arbitrarily high girth and with either many edges or large minimum degree. These subgraphs are created using probabilistic embeddings of a graph into extremal graphs.
KW - Degrees
KW - Forbidden subgraphs
KW - Girth
KW - Probabilistic methods
KW - Spanning subgraphs
UR - http://www.scopus.com/inward/record.url?scp=84925359639&partnerID=8YFLogxK
U2 - 10.1137/140954416
DO - 10.1137/140954416
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AN - SCOPUS:84925359639
SN - 0895-4801
VL - 29
SP - 65
EP - 78
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
IS - 1
ER -