The bipartite vertex (resp. edge) frustration of a graph G, denoted by ψ(G) (resp. φ(G)), is the smallest number of vertices (resp. edges) that have to be deleted from G to obtain a bipartite subgraph of G. A sharp lower bound of the bipartite vertex frustration of the line graph L(G) of every graph G is given. In addition, the exact value of ψ(L(G)) is calculated when G is a forest.

We consider the minimum feedback vertex set problem for some bipartite graphs and degree-constrained graphs. We show that the problem is linear time solvable for bipartite permutation graphs and NP-hard for grid intersection graphs. We also show that the problem is solvable in O(n2 log6 n) time for n-vertex graphs with maximum degree at most three. key words: 3-regular graph, bipartite permutat...

A graph U is (induced)-universal for a class of graphs X if every member of X is contained in U as an induced subgraph. We study the problem of finding a universal graph with minimum number of vertices for various classes of bipartite graphs: exponential classes of bipartite (and general) graphs, bipartite chain graphs, bipartite permutation graphs, and general bipartite graphs. For exponential...

Let G = (X, Y ) be a bipartite graph and define σ 2(G) = min{d(x) + d(y) : xy / ∈ E(G), x ∈ X, y ∈ Y }. Moon and Moser [5] showed that if G is a bipartite graph on 2n vertices such that σ 2(G) ≥ n + 1 then G is hamiltonian, sharpening a classical result of Ore [6] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that σ 2(G) ≥ n+ 2k− 1 then G contains k edge...

A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coalition graphs have been introduced and investigated in connection to perfect graphs. Recognizing and characterizing opposition and coalition graphs still remain lon...

For an undirected graph and a fixed integer k, a 2-matching is said to be Ck-free if it has no cycle of length k or less. In particular, a C4-free 2-matching in a bipartite graph is called a square-free 2-matching. The problem of finding a maximum Ck-free 2-matching in a bipartite graph is NP-hard when k ≥ 6, and polynomially solvable when k = 4. Also, the problem of finding a maximum-weight Ck...

In this paper, we deal with zero-divisor graphs of posets. We prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or ∞. We also characterize zero-divisor graphs of posets in terms of their diameter and girth. © 2012 Elsevier B.V. All rights reserved.

Any bipartite Eulerian graph, any Eulerian graph with evenly many vertices, and any bipartite graph with evenly many vertices and edges, has an even number of spanning trees. More generally, a graph has evenly many spanning trees if and only if it has an Eulerian edge cut. ∗Work supported in part by NSF grant CCR-9258355 and by matching funds from Xerox Corp.

A bipartite graph is chordal bipartite if every cycle of length at least six has a chord. In the class of chordal bipartite graphs the tree-width and the clique-width are unbounded. Our main results are that chordal bipartite graphs of bounded vertex degree have bounded tree-width and that k-fork-free chordal bipartite graphs have bounded clique-width, where a k-fork is the graph arising from a...

Our construction of error-correcting codes will exploit bipartite expander graphs (as these give a much cleaner construction than the general case). Let’s begin by examining what a bipartite expander graph should look like. It’s vertex set will have two parts, U and V , each having n vertices. Every vertex will have degree d, and every edge will go from a vertex in U to a vertex in V . In the s...