A Note on a Binary Relation Corresponding to a Bipartite Graph (original) (raw)

A note on the independence number in bipartite graphs

Australas. J Comb., 2016

The independence number of a graph G, denoted by α(G), is the maximum cardinality of an independent set of vertices in G. The transversal number of G is the minimum cardinality of a set of vertices that covers all the edges of G. If G is a bipartite graph of order n, then it is easy to see that n 2 ≤ α(G) ≤ n − 1. If G has no edges, then α(G) = n = n(G). Volkmann [Australas. J. Combin. 41 (2008), 219– 222] presented a constructive characterization of bipartite graphs G of order n for which α(G) = n 2 . In this paper we characterize all bipartite graphs G of order n with α(G) = k, for each n 2 ≤ k ≤ n − 1. We also give a characterization on the Nordhaus-Gaddum type inequalities on the transversal number of trees.

On some numerical characteristics of a bipartite graph

The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these characteristics of the set of all bipartite graphs of the type g=langleRgcupCg,Egrangleg=\langle R_g \cup C_g, E_g \rangleg=langleRgcupCg,Egrangle is formulated and proved, where V=RgcupCgV=R_g \cup C_gV=RgcupCg is the set of vertices, EgE_gEg is the set of edges of the graph ggg, $ |R_g |=m\ge 1$, ∣Cg∣=nge1|C_g |= n\ge 1Cg=nge1, ∣Eg∣=kge0|E_g |=k\ge 0Eg=kge0, m,nm,nm,n and kkk are integers.

Bipartite theory of irredundant set

Proyecciones (Antofagasta), 2011

The bipartite version of irredundant set, edge-vertex irredundant set and vertex-edge irredundant set are introduced. Using the bipartite theory of graph, IR ve (G) + γ(G) ≤ |V | and γ ve (G) + IR(G) ≤ |V | are proved.

On an Application of Bipartite Graphs

2012

Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general n 2 × n 2 case are discussed in this paper. All bipartite graphs of the type g = Rg ∪ Cg, Eg , where |Rg| = |Cg| = 2 or |Rg| = |Cg| = 3 are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the 4 × 4 and 9 × 9 cases are calculated.

SOME SPECIFIC CLASSES OF RELATIONS, A REVIEW

MAT-KOL, 2018

In this paper, the concepts of several new classes of relations on sets are presented introduced by this author in the previous five years. The following classes of relations have been introduced and partly described in several his articles: the class of quasi-regular, the class of quasi-conjugative, the class of quasi-normal and the class of normally conjugative relations.

Bipartite Graphs and Monochromatic Squares

Order, 2018

Let κ be a successor cardinal. We prove that consistently every bipartite graph of size κ + × κ + contains either an independent set or a clique of size τ × τ for every ordinal τ < κ +. We prove a similar theorem for ℓ-partite graphs. 2010 Mathematics Subject Classification. 05C63.

A pr 2 01 6 Canonical binary matrices related to bipartite graphs

2016

The current paper is dedicated to the problem of finding the number of mutually non isomorphic bipartite graphs of the type g = 〈Rg, Cg, Eg〉 at given n = |Rg | and m = |Cg |, where Rg and Cg are the two disjoint parts of the vertices of the graphs g, and Eg is the set of edges, Eg ⊆ Rg ×Cg. For this purpose, the concept of canonical binary matrix is introduced. The different canonical matrices unambiguously describe the different with exactness up to isomorphism bipartite graphs. We have found a necessary and sufficient condition an arbitrary matrix to be canonical. This condition could be the base for realizing recursive algorithm for finding all n×m canonical binary matrices and consequently for finding all with exactness up to isomorphism binary matrices with cardinality of each part equal to n and m.

Weshallusestandardgraphtheorynotation.Weconsideronlyfinite,undirected graphs.Allgraphswillbeassumedtohaveneitherloopsnormultipleedges. Let G = (L,R;E)beabipartitegraphwithapartition L, Randanedgeset

2006

A bipartite graph G = (L, R; E) where V (G) = L∪R, |L| = p, |R| = q is called a (p, q)-tree if |E(G)| = p + q − 1 and G has no cycles. A bipartite graph G = (L, R; E) is a subgraph of a bipartite graph H = (L′, R′; E′) if L ⊆ L′, R ⊆ R′ and E ⊆ E′. In this paper we present sufficient degree conditions for a bipartite graph to contain a (p, q)-tree.

Partition Relations and Transitivity Domains of Binary Relations

Journal of the London Mathematical Society, 1967

Introduction Let a, fi and y be order types and r a positive integer. The part&orb relation [I] a-48, r)' (1) expresses, by deli&ion, the following condition. If S is an ordered set, of order type tp A'= a, and if the set [S]" of all subsets of 8 of exactly r elements is arbitrarily expressed as the union of two sets K,, K,, then there always exists a set Xc X such that either tpX =p and [X~'C&,, or tpX=y and [X]' cK1. The following result is known [l ; Theorem 251 involving the least infinite ordinal wO = OJ and the negation a-t+ (/3, Y)? of (1). THEOREM 1.