On problems about judicious bipartitions of graphs

Abstract

Bollobás and Scott [5] conjectured that every graph $G$ has a balanced bipartite spanning subgraph $H$ such that for each $v\in V(G)$, $d_H(v)\ge (d_G(v)-1)/2$. In this paper, we show that every graphic sequence has a realization for which this Bollobás-Scott conjecture holds, confirming a conjecture of Hartke and Seacrest [10]. On the other hand, we give an infinite family of counterexamples to this Bollobás-Scott conjecture, which indicates that $\lfloor (d_G(v)-1)/2\rfloor$ (rather than $(d_G(v)-1)/2$) is probably the correct lower bound. We also study bipartitions $V_1, V_2$ of graphs with a fixed number of edges. We provide a (best possible) upper bound on $e(V_1)^λ+e(V_2)^λ$ for any real $λ\geq 1$ (the case $λ=2$ is a question of Scott [13]) and answer a question of Scott [13] on $\max\{e(V_1),e(V_2)\}$.