Bipartite subgraphs of triangle-free subcubic graphs

Abstract

Suppose G is a graph with n vertices and m edges. Let n ′ be the maximum number of vertices in an induced bipartite subgraph of G and let m ′ be the maximum number of edges in a spanning bipartite subgraph of G. Then b ( G ) = m ′ / m is called the bipartite density of G, and b ∗ ( G ) = n ′ / n is called the bipartite ratio of G. This paper proves that every 2-connected triangle-free subcubic graph, apart from seven exceptions, has b ( G ) ⩾ 17 / 21 . Every 2-connected triangle-free subcubic graph other than the Petersen graph and the dodecahedron has b ∗ ( G ) ⩾ 5 / 7 . The bounds that b ∗ ( G ) ⩾ 5 / 7 and b ( G ) ⩾ 17 / 21 are tight in the sense that there are infinitely many 2-connected triangle-free cubic graphs G for which b ( G ) = 17 / 21 and b ∗ ( G ) = 5 / 7 . On the other hand, if G is not cubic (i.e., G have vertices of degree at most 2), then the strict inequalities b ∗ ( G ) > 5 / 7 and b ( G ) > 17 / 21 hold, with a few exceptions. Nevertheless, the bounds are still sharp in the sense that for any ϵ > 0 , there are infinitely many 2-connected subcubic graphs G with minimum degree 2 such that b ∗ ( G ) < 5 / 7 + ϵ and b ( G ) < 17 / 21 + ϵ . The bound that b ( G ) ⩾ 17 / 21 is a common improvement of an earlier result of Bondy and Locke and a recent result of Xu and Yu: Bondy and Locke proved that every triangle-free cubic graph other than the Petersen graph and the dodecahedron has b ( G ) > 4 / 5 . Xu and Yu confirmed a conjecture of Bondy and Locke and proved that every 2-connected triangle free subcubic graph with minimum degree 2 apart from five exceptions has b ( G ) > 4 / 5 . The bound b ∗ ( G ) ⩾ 5 / 7 is a strengthening of a well-known result (first proved by Fajtlowicz and by Staton, and with a few new proofs found later) which says that any triangle-free subcubic graph G has independence ratio at least 5/14. The proofs imply a linear time algorithm that finds an induced bipartite subgraph H of G with | V ( H ) | / | V ( G ) | ⩾ 5 / 7 and a spanning bipartite subgraph H ′ of G with | E ( H ′ ) | / | E ( G ) | ⩾ 17 / 21 .