A note on balanced bipartitions

14Citations
Citations of this article
8Readers
Mendeley users who have this article in their library.

Abstract

A balanced bipartition of a graph G is a bipartition V 1 and V 2 of V(G) such that -1≤|V 1|-|V 2|≤1. Bollobs and Scott conjectured that if G is a graph with m edges and minimum degree at least 2 then G admits a balanced bipartition V 1,V 2 such that maxe(V 1),e(V 2)≤m3, 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 Δ(G)=o(n) or the minimum degree δ(G)→∞, then G admits a balanced bipartition V 1,V 2 such that maxe(V 1),e(V 2) ≤(1+o(1))m4, answering a question of Bollobs and Scott in the affirmative. We also provide a sharp lower bound on maxe(V 1,V 2):V 1,V 2 is a balanced bipartition of G, in terms of size of a maximum matching, where e(V 1,V 2) denotes the number of edges between V 1 and V 2. © 2010 Elsevier B.V.

Cite

CITATION STYLE

APA

Xu, B., Yan, J., & Yu, X. (2010). A note on balanced bipartitions. Discrete Mathematics, 310(20), 2613–2617. https://doi.org/10.1016/j.disc.2010.03.029

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free