Choosability with Separation of Complete Multipartite Graphs and Hypergraphs

Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

A cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form {F, F , F θ 2 , . . . , F θ q−1 } for some permutation θ of the vertex set V . Let Vn = {1, 2, . . . , n}. For a positive integer k, ( Vn k ) denotes the set of all k-subsets of Vn. For a nonempty subset K of Vn−1, we let K n denote the hypergraph ( Vn, ⋃ k∈K ( Vn k )) . In this paper, we find a necessary an...

On graphs with complete multipartite µ-graphs

Integral complete multipartite graphs

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphsKp1,p2,...,pr =Ka1·p1,a2·p2,...,as ·ps with s=3, 4.We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s = 4, we giv...

Decompositions of complete multipartite graphs

This paper answers a recent question of Dobson and Marušič by partitioning the edge set of a complete bipartite graph into two parts, both of which are edge sets of arctransitive graphs, one primitive and the other imprimitive. The first member of the infinite family is the one constructed by Dobson and Marušič.

On Choosability with Separation of Planar Graphs with Forbidden Cycles

We study choosability with separation which is a constrained version of list coloring of graphs. A (k, d)-list assignment L of a graph G is a function that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k, d)-choosable if there exists an L-coloring of G for every (k, d)-list assignment L. This...