We show that for any 2-local colouring of the edges of a complete bipartite graph, its vertices can be covered with at most 3 disjoint monochromatic paths. And, we can cover almost all vertices of any complete or complete bipartite r-locally coloured graph with O(r) disjoint monochromatic cycles. MSC: 05C38, 05C55.

We shall consider the problem of embedding the complete bipartite graph, Km,n, onto a linear and cyclic chassis in such a way as to minimize the cutwidth. The linear cutwidth of the complete bipartite graph is established and a partial solution to the cyclic cutwidth is presented. It is known that there is a paper in existance [3] that has established the linear cutwidth of the complete biparti...

Antipodal covers of strongly regular graphs which are not necessarily distance-regular are studied. The structure of short cycles in an antipodal cover is considered. In most cases, this provides a tool to determine if a strongly regular graph has an antipodal cover. In these cases, covers cannot be distance-regular except when they cover a complete bipartite graph. A relationship between antip...

A graph with signed edges is orientation embedded in a surface when it is topologically embedded so that one trip around a closed path preserves or reverses orientation according as the path's sign product is positive or negative. We nd the smallest surface within which it is possible to orientation-embed the complete bipartite signed graph K r;s , which is obtained from the complete bipartite ...

To minimize the access cost in large disk arrays (RAID) Cohen et al. [5–7] introduced (d, f)-cluttered orderings of various set systems, d, f ∈ N. In case of a graph this amounts to an ordering of the edge set such that the number of points contained in any d consecutive edges is bounded by the number f . For the complete graph, Cohen et al. gave some optimal solution for small parameters d [5]...

The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application o...

G(n) denotes a graph of n vertices and G(n) denotes its complementary graph. In a complete graph every two distinct vertices are joined by an edge. Let C k (G(n)) denote the number of complete subgraphs of k vertices contained in G(n). Recently it was proved [1] that for every k 2 (n) (1) min C (G (n)) + Ck(G(n)) < k k, , ! 2 2 where the minimum is over all graphs G(n). It seems likely that (1)...

We give an exponential lower bound for the Graver complexity of the incidence matrix of a complete bipartite graph of arbitrary size. Our result is a generalization of the result by Berstein and Onn [2] for the complete bipartite graph K3,r, r ≥ 3.

It is well-known that the GRAPH 3.COLORABILITY problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in (1) a tree or (2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given gr...

A graph is polar if the vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete proble...