We first consider the following problem. We are given a fixed perfect matching M of [n] and we add random edges one at a time until there is a Hamilton cycle containing M . We show that w.h.p. the hitting time for this event is the same as that for the first time there are no isolated vertices in the graph induced by the random edges. We then use this result for the following problem. We genera...

It is shown that for every > 0 and n > n0( ), any complete graph K on n vertices whose edges are colored so that no vertex is incident with more than (1 − 1 √ 2 − )n edges of the same color, contains a Hamilton cycle in which adjacent edges have distinct colors. Moreover, for every k between 3 and n any such K contains a cycle of length k in which adjacent edges have distinct colors.

Let Kc n denote a complete graph on n vertices whose edges are colored in an arbitrary way. Let ∆mon(Kc n) denote the maximum number of edges of the same color incident with a vertex of Kn. A properly colored cycle (path) in Kc n is a cycle (path) in which adjacent edges have distinct colors. B. Bollobás and P. Erdös (1976) proposed the following conjecture: If ∆mon(Kc n) < b 2 c, then Kc n con...

A combinatorial bijection between k-edge colored trees and colored Prüfer codes for labelled trees is established. This bijection gives a simple combinatorial proof for the number k(n − 2)!(nk−n n−2 ) of k-edge colored trees with n vertices.

Nevus lipomatosus cutaneous superficialis is a rare idiopathic hamartomatous anomaly of the skin characterized by the presence of clusters of mature fat cells among the collagen bundles of the dermis. The classic, or solitary type, presents with asymptomatic soft, skin-colored to yellow papules or nodules. We report the case of a 12-year-old boy with congenital, classical nevus lipomatosus cuta...

A (m,n)-colored mixed graph is a graph having arcs of m different colors and edges of n different colors. A graph homomorphism of a (m,n)-colored mixed graph G to a (m,n)colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is also an arc (edge) of color c. The (m,n)-colored mixed chromatic number χ(m,n)(G) of a (m,n)-colored mixed graph G, in...

The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this approach we introduce colored analogs of P -partitions and enriched P -partitions. We also frame our results in terms of Aguiar, Bergeron, and Sottile’s the...

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