Assume that G = G(V, E) is an undirected graph with vertex set V and edge set E. A clique of G is a complete subgraph. An edge clique-covering is a family of cliques of G which cover all edges of G. The edge clique-cover number, Be( G), is the minimum number of cliques in an edge clique-cover of G. For results and applications of the edge clique-cover number see [1-4]. Observe that Be(G) does n...

Online Bipartite Matching is a generalization of a well-known Bipartite Matching problem. In a Bipartite Matching, we a given a bipartite graph G = (L,R,E), and we need to find a matching M ⊆ E such that no edges in M have common endpoints. In the online version L is known, but vertices in R are arriving one at a time. When vertex j ∈ R arrives (with all its edges), we need to make an irreversi...

A graph is edge-distance-regular when it is distance-regular around each of its edges and it has the same intersection numbers for any edge taken as a root. In this paper we give some (combinatorial and algebraic) proofs of the fact that every edge-distance-regular graph Γ is distance-regular and homogeneous. More precisely, Γ is edge-distance-regular if and only if it is bipartite distance-reg...

Abstract The standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms discrete path integration on planar bicoloured (plabic) graphs disc. An alternative proposed T. Lam [38] systems relations at vertices such graphs, depending some signatures defined their edges. problem characterizing corresponding to ce...

Fouquet, J.-L. and A.P. Wojda, Mutual placement of bipartite graphs, Discrete Mathematics 121 (1993) 85-92. Let G =(L, R, E) and H =(L’, R’, E’) be bipartite graphs. A bijection 4: L w R + L’v R’ is said to be a biplacement of G and H if 4(L)= L’ and ~$(x)~$(y)gE’ f or every edge xy of G. A biplacement of G and its copy is called a 2-placement of G. We prove that, with some exceptions, every bi...

This thesis focuses on three fundamental problems in combinatorial optimization: non-bipartite matching, matroid intersection, and submodular function minimization. We develop simple, efficient, randomized algorithms for the first two problems, and prove new lower bounds for the last two problems. For the matching problem, we give an algorithm for constructing perfect or maximum cardinality mat...

As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekulé structures (1-factors). A bipartite graph G is called elementary if G is connected and every edge belongs to a 1-factor of G. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given. A peripheral face f of a plane elementary graph is reducible, if th...

We study the online matroid intersection problem, which is related to the well-studied online bipartite matching problem in the vertex arrival model. For two matroids M1 and M2 defined on the same ground set E, the problem is to design an algorithm that constructs a large common independent set in an online fashion. The algorithm is presented with the ground set elements one-by-one in a uniform...

Let Γ denote a bipartite distance-regular graph with diameter D. In [J. S. Caughman, Bipartite Q-polynomial distance-regular graphs, Graphs Combin. 20 (2004), 47–57], Caughman showed that if D > 12, then Γ is Q-polynomial if and only if one of the following (i)-(iv) holds: (i) Γ is the ordinary 2D-cycle, (ii) Γ is the Hamming cube H(D, 2), (iii) Γ is the antipodal quotient of the Hamming cube H...