We show that for any colouring of the edges of the complete bipartite graphKn,n with 3 colours there are 5 disjoint monochromatic cycles which together cover all but o(n) of the vertices and 18 disjoint monochromatic cycles which together cover all vertices. keywords: monochromatic cycle partition, Ramsey-type problem, complete bipartite graph MSC 2010: 05C69, 05C75, 05C38.

We prove that the maximum edge biclique problem in bipartite graphs is NP-complete. A biclique in a bipartite graph is a vertex induced subgraph which is complete. The problem of finding a biclique with a maximum number of vertices is known to be solvable in polynomial time but the complexity of finding a biclique with a maximum number of edges was still undecided.

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.

The design of large disk array architectures leads to interesting combinatorial problems. Minimizing the number of disk operations when writing to consecutive disks leads to the concept of “cluttered orderings” which were introduced for the complete graph by Cohen et al. (2001). Mueller et al. (2005) adapted the concept of wrapped Δ-labellings to the complete bipartite case. In this paper, we g...

A graph G is hereditarily dominated by a class D of connected graphs if each connected induced subgraph of G contains a dominating induced subgraph belonging to D. In this paper we characterize graphs hereditarily dominated by classes of complete bipartite graphs, stars, connected bipartite graphs, and complete k-partite graphs.

A series of recent papers shows that it is NP-complete to decide whether an oriented graph admits a homomorphism to the tournament T4 on 4 vertices containing a 4-circuit, each time on a smaller graph class. We improve these results by showing that homomorphism to T4 is NP-complete for bipartite planar subcubic graphs of arbitrarily large fixed girth. We also show that push homomorphism is NP-c...

A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...

Testing if a given graph G contains the k-vertex path Pk as a minor or as an induced minor is trivial for every fixed integer k ≥ 1. However, the situation changes for the problem of checking if a graph can be modified into Pk by using only edge contractions. In this case the problem is known to be NP-complete even if k = 4. This led to an intensive investigation for testing contractibility on ...

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 ...

We previously looked at communities as simply dense subgraphs graphs, or as subgraphs such that each node has a large fraction of its edges inside. Another view, taken in [3], starts from the hubs and authorities model. It argues that a structure of densely linked hubs and authorities is a common feature of communities. Such a core, a dense bipartite graph, can be considered the “signature” of ...