Perfect k-Colored Matchings and $$(k+2)$$ ( k + 2 ) -Gonal Tilings

Perfect Fractional Matchings in $k$-Out Hypergraphs

Extending the notion of (random) k-out graphs, we consider when the k-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each r there is a k = k(r) such that the k-out r-uniform hypergraph on n vertices has a perfect fractional matching with high probability (i.e., with probability tending to 1 as n → ∞) and prove an analogous result for r-uniform r-...

PERFECT MATCHINGS AND k-DECOMPOSABILITY OF INCREASING TREES

A tree is called k-decomposable if it has a spanning forest whose components are all of size k. In this paper, we study the number of k-decomposable trees in families of increasing trees, i.e. labeled trees in which the unique path from the root to an arbitrary vertex forms an increasing sequence. Functional equations for the corresponding counting series are provided, yielding asymptotic or ev...

A classification of outerplanar K-gonal 2-trees

Essentially, a 2-tree is a simple connected graph composed by triangles glued along their edges in a tree-like fashion, that is, without cycles (of triangles). The enumeration of 2-trees has been extensively studied in the literature. See, for instance, Harary and Palmer [8] and Fowler et al [6]. Here we consider more general 2-trees, where the triangles are replaced by quadrilaterals, pentagon...

Bipartite, k-Colorable and k-Colored Graphs

Perfect Matchings, Tilings and Hamilton Cycles in Hypergraphs

This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite Hajnal-Szemerédi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs....