α_{2}-labeling of graphs

We show that if a graph $G$ on $n$ edges allows certain special type of rosy labeling (a.k.a. $\rho$-labeling), called $\alpha_2$-labeling, then for any positive integer $k$ the complete graph $K_{2nk+1}$ can be decomposed into copies of $G$. This notion generalizes the $\alpha$-labeling introduced in 1967 by A. Rosa

Factorizations of complete graphs into brooms

Let r and n be positive integers with r<2n. A broom of order 2n is the union of the path on P2n−r−1 and the star K1,r, plus one edge joining the center of the star to an endpoint of the path. It was shown by Kubesa (2005) [10] that the broom factorizes the complete graph K2n for odd n and View the MathML source. In this note we give a complete classification of brooms that factorize K2n by giving a constructive proof for all View the MathML source (with one exceptional case) and by showing that the brooms for View the MathML source do not factorize the complete graph K2n.Web of Science31261093108

Decomposition of complete bipartite graphs into generalized prisms

R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K6n,6n. In (Cichacz and Fronček, 2009) [2] the first two authors established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph Kn,n into certain families of 3-regular graphs of order 2n. In this paper we tackle the problem of decompositions of Kn,n into certain 3-regular graphs called generalized prisms. We will show that certain families of 3-regular graphs of order 2n decompose the complete bipartite graph View the MathML source.Web of Science34111010

Factorizations of complete graphs into caterpillars of diameter 5

We examine factorizations of complete graphs K2n into caterpillars of diameter 5. First we present a construction generalizing some previously known methods. Then we use the new method along with some previous partial results to give a complete characterization of caterpillars of diameter 5, which factorize the complete graph K2n

Factorizations of complete graphs into trees with at most four non-leave vertices

We give a complete characterization of trees with at most four non-leave vertices, which factorize the complete graph K 2n

Decompositions of complete multipartite graphs into disconnected selfcomplementary factors

We determine the spectrum of complete bipartite and tripartite graphs that are decomposable into disconnected selfcomplementary factors (isodecomposable). For r-partite graphs with r greater than or equal to 4 we determine the smallest orders of graphs that are isodecomposable. We also prove that every complete r-partite graph with at least one even part is isodecomposable. For graphs with all odd parts we prove that if among the cardinalities of the parts there is exactly one that appears an odd number of times, then the graph is also isodecomposable. Finally, we present a class of graphs with all odd parts that are not isodecomposable

Disconnected self-complementary factors of almost complete tripartite graphs

A complete tripartite graph without one edge, (K) over tilde(m1,m2,m3), is called an almost complete tripartite graph. A graph (K) over tilde(m1,m2,m3) that can be decomposed into two isomorphic factors is called halvable. It is proved that an almost complete tripartite graph is halvable into disconnected factors without isolated vertices if an only if it is a graph (K) over tilde(1,2m+1,2p) and the "missing" (i.e., deleted) edge has the endvertices in the odd parts. It is also shown that the factors have always two components: one component is isomorphic to a star K-1,K-p, and the other to a graph K-1,K-2m,K-p - K-1,K-m. For factors with isolated vertices it is proved that they have just one non-trivial component and all isolated vertices belong to the same part