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TWO PROBLEMS ON BIPARTITE GRAPHS
, 2009
"... Erdös proved that every graph G has a bipartite, spanning subgraph B such that dB(v) ≥ dG(v) 2 for any v ∈ V (G). Bollobás and Scott conjectured that every graph G has a balanced, bipartite, spanning subgraph B such that dB(v) ≥ dG(v)−1 2. We prove this for graphs with maximum degree 3. However, th ..."
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Erdös proved that every graph G has a bipartite, spanning subgraph B such that dB(v) ≥ dG(v) 2 for any v ∈ V (G). Bollobás and Scott conjectured that every graph G has a balanced, bipartite, spanning subgraph B such that dB(v) ≥ dG(v)−1 2. We prove this for graphs with maximum degree 3. However, the majority of this paper is focused on bipartite graph tiling. We prove a conjecture of Zhao that implies an asymptotic version Kühn and Osthus ' tiling result when restricted to a bipartite graph H. Speci cally, we prove for any bipartite graph H on h vertices, if G is a bipartite graph on 2n = mh vertices and δ(G) ≥ (1 − 1 χ ∗)n + γn, then G (H) contains an Htiling where χ ∗ (H) is either the chromatic number or the critical chromatic number of H.
A note on balanced bipartitions
"... A balanced bipartition of a graph G is a bipartition V1 and V2 of V (G) such that −1 ≤ V1  − V2  ≤ 1. Bollobás and Scott conjectured that if G is a graph with m edges and minimum degree at least 2 then G admits a balanced bipartition V1, V2 such that max{e(V1), e(V2)} ≤ m/3, where e(Vi) denot ..."
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A balanced bipartition of a graph G is a bipartition V1 and V2 of V (G) such that −1 ≤ V1  − V2  ≤ 1. Bollobás and Scott conjectured that if G is a graph with m edges and minimum degree at least 2 then G admits a balanced bipartition V1, V2 such that max{e(V1), e(V2)} ≤ m/3, where e(Vi) denotes the number of edges of G with both ends in Vi. In this note, we prove this conjecture for graphs with average degree at least 6 or with minimum degree at least 5. Moreover, we show that if G is a graph with m edges and n vertices, and if the maximum degree ∆(G) = o(n) or the minimum degree δ(G) → ∞, then G admits a balanced bipartition V1, V2 such that max{e(V1), e(V2)} ≤ (1 +o(1))m/4, answering a question of Bollobás and Scott in the affirmative. We also provide a sharp lower bound on max{e(V1, V2) : V1, V2 is a balanced bipartition of G}, in terms of size of a maximum matching, where e(V1, V2) denotes the number of edges between V1 and V2.