Atif A. Abueida

Multidesigns for Graph-Pairs of Order 4 and 5

AbstractThe graph decomposition problem is well known. We say a subgraph GdividesKm if the edges of Km can be partitioned into copies of G. Such a partition is called a G-decomposition or G-design. The graph multidecomposition problem is a variation of the above. By a graph-pair of ordert, we mean two non-isomorphic graphs G and H on t non-isolated vertices for which G∪H≅Kt for some integer t≥4. Given a graph-pair (G,H), if the edges of Km can be partitioned into copies of G and H with at least one copy of G and one copy of H, we say (G,H) divides Km. We will refer to this partition as a (G,H)-multidecomposition. In this paper, we consider the existence of multidecompositions for several graph-pairs. For the pairs (G,H) which satisfy G∪H≅K4 or K5, we completely determine the values of m for which Km admits a (G,H)-multidecomposition. When Km does not admit a (G,H)-multidecomposition, we instead find a maximum multipacking and a minimum multicovering. A multidesign is a multidecomposition, a maximum multipacking, or a minimum multicovering.

A Min–Max Property of Chordal Bipartite Graphs with Applications

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced matching in G, and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph.

Multidecompositions of Several Graph Products

We find necessary and sufficient conditions for (C4, E2) multidecompositions of the cartesian product and tensor product of paths, cycles, and complete graphs.

On the Decompositions of Complete Graphs into Cycles and Stars on the Same Number of Edges

Abstract Let Cm and Sm denote a cycle and a star on m edges, respectively. We investigate the decomposition of the complete graphs, Kn, into cycles and stars on the same number of edges. We give an algorithm that determines values of n, for a given value of m, where Kn is {Cm, Sm}-decomposable. We show that the obvious necessary condition is sufficient for such decompositions to exist for different values of m.

Note: A note on the recognition of bisplit graphs

We show that bisplit graphs can be recognized in O(n^2) time. The previous best bound of O(mn) for the problem appeared in a recently published article [A. Brandstadt, P.L. Hammer, V.B. Le, V.V. Lozin, Bisplit graphs, Discrete Math. 299 (2005) 11-32] in this journal.

Completing a solution of the embedding problem for incomplete idempotent latin squares when numerical conditions suffice

Abstract Building upon the work of several previous papers, necessary and sufficient conditions are obtained for an incomplete idempotent latin square R of order n to be embedded in an idempotent latin square of order 2 n . This work extends known necessary and sufficient conditions for embeddings into idempotent latin squares of order t ≥ 2 n + 1 , thereby pushing the bounds down to the point where subsequent developments must handle additional conditions that involve the arrangement of symbols on the given square.

INVERSE SPREAD LIMIT OF A NONNEGATIVE MATRIX

For a given nonnegative n × n matrix A consider the following quantity as long as the denominator is positive. It is simply the ratio between the smallest and the largest entries of Am. We call s(Am) the inverse spread of Am which is interpreted as a measure of the maximum variation among the entries of Am in the multiplicative and reciprocal sense. Smaller s(Am) means a larger variation for Am. Clearly 0 = s(Am) = 1 for all m = 1, 2, . . . We study the asymptotic behavior of s(Am), that is, the behavior of s(Am) as m ? 8. The study arises from evolutionary biology.