We introduce a search problem for finding a regular bi-partite graph of maximum attainable girth for specified degree and number of vertices, by restricting the search space using a series of mathematically rigourous arguments from [1] and [2]. The goal of this paper is to derive the enumeration search algorithm for finding a girth maximum (m, r) BTU, which is notation for regular partite graph...

Let r ≥ 2 and c > 0. Every graph on n vertices with at least cnr cliques on r vertices contains a complete r-partite subgraph with r − 1 parts of size ⌊cr log n⌋ and one part of size greater than n1−c r−1 . This result implies the Erdős-Stone-Bollobás theorem, the essential quantitative form of the Erdős-Stone theorem.

We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 in which this upper bound is attained, this graph can be viewed as a generalization of the ...

Transversals in r-partite graphs with various properties are known to have many applications in graph theory and theoretical computer science. We investigate f -bounded transversals (or f -BT), that is, transversals whose connected components have order at most f . In some sense we search for the the sparsest f -BT-free graphs. We obtain estimates on the smallest maximum degree that 3-partite a...

A signed k-partite graph (signed multipartite graph) is a k-partite graph in which each edge is assigned a positive or a negative sign. If G(V1, V2, · · · , Vk) is a signed k-partite graph with Vi = {vi1, vi2, · · · , vini}, 1 ≤ i ≤ k, the signed degree of vij is sdeg(vij) = dij = d + ij − d − ij , where 1 ≤ i ≤ k, 1 ≤ j ≤ ni and d + ij(d − ij) is the number of positive (negative) edges inciden...

In this paper we prove an asymptotic multipartite version of a well-known theorem of Kühn and Osthus by establishing, for any graph H with chromatic number r, the asymptotic multipartite minimum degree threshold which ensures that a large r-partite graph G admits a perfect H-tiling. We also give the threshold for an H-tiling covering all but a linear number of vertices of G, in a multipartite a...

We call a graph G O-graph if there is an optimal coloring of the set of vertices and an optimal (disjoint) covering with cliques such that any class of colors intersects any clique. In this paper, it has been established the relation to [p, q, r]-partite graphs and the fact that the O-graphs admit a k-clique-join.

We extend results on monochromatic tree covers and from classical Ramsey theory to a generalised setting, where each of the edges of an underlying host graph (here, either a complete graph or a complete bipartite graph), is coloured with a set of colours. Our results for tree covers in this setting have an application to Ryser’s Conjecture. Every r-partite r-uniform hypergraph whose edges pairw...

Abstract The book graph $B_n ^{(k)}$ consists of $n$ copies $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ Here consider natural off-diagonal variant $r(B_{cn} B_n^{(k)})$ for fixed $c \in (0,1]$ more general setting, show that an interesting dichotomy emerges: very small $c$ , simple $k$ -partite construction ...

A sequence of nonnegative integers π is graphic if it is the degree sequence of some graph G. In this case we say that G is a realization of π, and we write π = π(G). A graphic sequence π is potentially H-graphic if there is a realization of π that contains H as a subgraph. Given nonincreasing graphic sequences π1 = (d1, . . . , dn) and π2 = (s1, . . . , sn), we say that π1 majorizes π2 if di ≥...