Definition (Maximal size complete bipartite induced subgraph). β(G) := size of maximal complete bipartite induced subgraph of G. Definition (Minimal bipartite decomposion number). τ(G) := minimal number of complete edge disjoint covering bipartite subgraphs of G. Definition (Minimal nontrivial bipartite decomposion number). τ (G) := minimal number of complete edge disjoint covering nontrivial (...

It is shown that if G is any bipartite 2-regular graph of order at most n2 or at least n − 2, then the obvious necessary conditions are sufficient for the existence of a decomposition of the complete graph of order n into a perfect matching and edge-disjoint copies of G.

A formalization of simple graphs is given from scratch. There is already an article [10], dealing with the similar subject. It is not used as a starting-point, because [10] formalizes directed non-empty graphs. Given a set of vertices, edge is defined as an (unordered) pair of different two vertices and graph as a pair of a set of vertices and a set of edges. The following concepts are introduc...

In this paper, we present a linear time recognition algorithm for recognizing bipartite graphs without induced subgraphs isomorph to star123. Bipartite star123-free graphs are a natural generalization of both weak bisplit and Star123, Sun4free bipartite graphs, both further generalizing bicographs.

An orthogonal double cover (ODC) of the complete graph is a collection of graphs such that every two of them share exactly one edge and every edge of the complete graph belongs to exactly two of the graphs. In this paper, we consider the case where the graph to be covered twice is the complete bipartite graph Kmn,mn (for any values ofm,n) and all graphs in the collection are isomorphic to certa...

Structural properties of isometric subgraphs of Hamming graphs are presented, generalizing certain results on quasi-median graphs. Consequently, a relation on the edge set of a graph which is closely related to Winkler–Djokovi6 c’s relation is introduced and used for a characterization of isometric subgraphs of Hamming graphs. Moreover, some results considering semi-median graphs and expansions...

A 1978 theorem of Kozen states that two graphs on n vertices are isomorphic if and only if there is a clique of size n in the weak modular product between the two graphs. Restricting to bipartite graphs and considering complete bipartite subgraphs (bicliques) therein, we study the combinatorics of the weak modular product. We identify cases where isomorphism is tractable using this approach, wh...

the concept of the bipartite divisor graph for integer subsets has been considered in [‎graph combinator., ‎ 26 (2010) 95--105‎.]. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g$ is a solvable group‎, ‎then the number of connected components of this graph is at most $2$ and if $g...