Largest bipartite subgraphs in triangle‐free graphs with maximum degree three

Abstract

Let G be a triangle-free, loopless graph with maximum degree three. We display a polynomi$ algorithm which returns a bipartite subgraph of G containing at least 5 of the edges of G. Furthermore, we characterize the dodecahedron and the Petersen graph as the only 3-regular, triangle-free, loopless, connected graphs for which no bipartite subgraph has more than this proportion.

Maximum Directed Cuts in Graphs with Degree Constraints

“…We then give an alternative proof of the result of Lehel et al [13] by performing a simple operation on digraphs and applying a result of Bondy and Locke [4]. We also show that there are infinitely many rational numbers in the interval [1/3, 2/5] which could serve as the dicut density of a digraph in D (1,1).…”

“…, and let To characterize the irreducible digraphs in D that have dicut density 2/5, we also need the following result which was conjectured by Bondy and Locke [4], and proved recently by the present authors [16]. [16]) If the graph G is triangle-free and subcubic, and if each maximum cut of G has exactly (4/5)ε(G) edges, then G is one of the graphs in Figure 3.…”

“…We need a result of Bondy and Locke [4] on max cuts in triangle-free subcubic graphs. [4]) If G is a triangle-free subcubic graph, then G has a cut of size at least (4/5)ε(G), and such a cut can be found in O(|V (G)| 2 ) time.…”

“…[4]) If G is a triangle-free subcubic graph, then G has a cut of size at least (4/5)ε(G), and such a cut can be found in O(|V (G)| 2 ) time.…”

Searching maximum quasi-bicliques from protein-protein interaction network

“…Although many researchers [7,8,9,10,11] have developed some algorithms to solve the maximum biclique problem, they often focused on some special characteristics of the graph, so the problem is still intractable. Therefore, in computational biology field, some researchers mined quasi-bicliques instead of exact bicliques.…”