Let the Andrásfai graph Andk be defined as the graph with vertex set {v0, v1, . . . , v3k−2} and two vertices vi and vj being adjacent iff |i − j| ≡ 1 mod 3. The graphs Andk are maximal triangle-free and play a role in characterizing triangle-free graphs with large minimum degree as homomorphic preimages. A minimal bipartification of a graph G is defined as a set of edges F ⊂ E(G) having the pr...

We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U, V,E) and a parameter β, the goal is to find a subset V ′ ⊆ V containing β fraction of the vertices of V which minimizes the size of N(V ′), the neighborhood of V ′. This problem, which we call Bipartite Expansion, is a special case of sub...

For a fixed target graph H , the minimum cost homomorphism problem, MinHOM(H), asks, for a given graph Gwith integer costs ci(u), u ∈ V (G), i ∈ V (H), and an integer k, whether or not there exists a homomorphism of G to H of cost not exceeding k. When the target graph H is a bipartite graph a dichotomy classification is known: MinHOM(H) is solvable in polynomial time if and only if H does not ...

In a breakthrough work, Marcus et al. [15] recently showed that every d-regular bipartite Ramanujan graph has a 2-lift that is also d-regular bipartite Ramanujan. As a consequence, a straightforward iterative brute-force search algorithm leads to the construction of a d-regular bipartite Ramanujan graph on N vertices in time 2. Shift k-lifts studied in [1] lead to a natural approach for constru...

In a breakthrough work, Marcus et al. [15] recently showed that every d-regular bipartite Ramanujan graph has a 2-lift that is also d-regular bipartite Ramanujan. As a consequence, a straightforward iterative brute-force search algorithm leads to the construction of a d-regular bipartite Ramanujan graph on N vertices in time 2. Shift k-lifts studied in [1] lead to a natural approach for constru...

A graph is polar if the vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete proble...

A graph of order n is p-factor-critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1-factor-critical graphs and 2-factor-critical graphs are well-known factor-critical graphs and bicritical graphs, respectively. It is known that if a connected vertextransitive graph has odd order, then it is factor-critical...

Böttcher, Schacht and Taraz [6] gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós [15]. We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence...

A k-container C(u,v) of a graph G is a set of k disjoint paths joining u to v. A k-container C(u,v) is a k∗-container if every vertex of G is incident with a path in C(u,v). A bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u, v from different partite set of G. A bipartite graph G with connectivity k is super laceable if it is i∗-laceable for all i k. A ...

Borowiecki, M. and E. Drgas-Burchardt, Classes of chromatically unique graphs, Discrete Mathematics Ill (1993) 71-75. We prove that graphs obtained from complete equibipartite graphs by deleting some independent sets of edges are chromatically unique. 1. Preliminary definitions and results In this paper we consider finite, undirected, simple and loopless graphs. Two graphs G and H are said to b...