Bipartite regulation numbers (original) (raw)

On the regulation number of a multigraph

Graphs and Combinatorics, 1985

The regulation number of a multigraph G having maximum degree d is the minimum number of additional vertices that are required to construct a d-regular supermultigraph of G. It is shown that the regulation number of any multigraph is at most 3. The regulation number of a multidigraph is defined analogously and is shown never to exceed 2. A multigraph G has strength m if every two distinct vertices of G are joined by at most m parallel edges. For a multigraph G of strength m and maximum degree d, the m-regulation number of G is the minimum number of additional vertices that are required to construct a d-regular supermultigraph of G having strength m. A sharp upper bound on the 2-regulation number of a multigraph is shown to be (d + 5)/2, and a conjecture for general m is presented.

On bipartite graphs of defect at most 4

Discrete Applied Mathematics, 2010

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers ∆ ≥ 2 and D ≥ 2, find the maximum number N b (∆, D) of vertices in a bipartite graph of maximum degree ∆ and diameter D. In this context, the Moore bipartite bound M b (∆, D)

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 1∣Cg∣=nge1, ∣Eg∣=kge0|E_g |=k\ge 0∣Eg∣=kge0, m,nm,nm,n and kkk are integers.

On bipartite biregular large graphs

arXiv (Cornell University), 2024

A bipartite graph G = (V, E) with V = V 1 ∪V 2 is biregular if all the vertices of each stable set, V 1 and V 2 , have the same degree, r and s, respectively. This paper studies difference sets derived from both Abelian and non-Abelian groups. From them, we propose some constructions of bipartite biregular graphs with diameter d = 3 and asymptotically optimal order for given degrees r and s. Moreover, we find some biMoore graphs, that is, bipartite biregular graphs that attain the Moore bound.

Topological minors in bipartite graphs

Acta Mathematica Sinica, English Series, 2011

For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K (s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem. C., et al. ex(n, T K p ) which represents the maximum number of edges of a graph on n vertices free of a topological minor T K p of a complete graph on p vertices (see Bollobás' excellent monograph devoted to this subject and the contributions on this topic ). The second was stated by Zarankiewicz [10] who studied the maximum size of a bipartite graph on (m, n) vertices, denoted by z(m, n; s, t) that contains no bipartite complete K (s,t) subgraph with s vertices in the m-class and t vertices in the n-class. For a survey of this problem we also refer the reader to Section VI.2 of the book by Bollobás [3]. Most of the contributions are bounds for the function z(m, n; s, t) when s, t are fixed and m, n are much larger than s, t [11-13]. Other contributions provide exact values of the extremal function [14-16]. Recent results on some problems involving the existence of a complete bipartite graph or a subdivision of a complete bipartite graph as a subgraph can be found in the literature [17-20]. Böhme et al. [21] studied the size of a k-connected graph free of either an induced path of a given length or a subdivision of a complete bipartite graph. Kühn and Osthus [22] proved that for any graph H and for every integer s there exists a function f = f (H, s) such that every graph of size at least f contains either a K s,s as a subgraph or an induced subdivision of H. Meyer [17] also relates the size of a graph with the property of containing a minor of K s,t .

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.

Some results on placing bipartite graphs of maximum degree two

Electronic Notes in Discrete Mathematics, 2006

We prove that if G and H are two bipartite graphs of order |G| = |H| = 2p ≥ 4 such that the sizes of G and H satisfy G ≤ 2p − 3 and H ≤ 2p − 2, and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs. This result implies easily that for integers p and k 1 , k 2 ,. .. , k l such that k i ≥ 2 for i = 1,. .. , l and k 1 +. .. + k l ≤ p − 1 every bipartite balanced graph G of order 2p and size at least p 2 − 2p + 3 contains mutually vertex disjoint cycles C 2k 1 ,. .. , C 2k l , unless G = K 3,3 − 3K 1,1 .

Constructions of small regular bipartite graphs of girth 6

2011

In this article, some structures in the projective plane of order q are found which allow us to construct small kregular balanced bipartite graphs of girth 6 for all k ≤ q. When k = q, the order of these q-regular graphs is 2(q 2 −1); and when k ≤ q −1, the order of these k -regular graphs is 2(qk − 2). Moreover, the incidence matrix of a k -regular balanced bipartite graph of girth 6 having 2(qk − 2) vertices, where k is an integer and q is a prime power with 3 ≤ k ≤ q − 1, is provided. These graphs improve upon the best known upper bounds for the number of vertices in regular graphs of girth 6.