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

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

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

Self-complementary factors of almost complete tripartite graphs of even order

A complete tripartite graph without one edge, (K) over tilde (m1,m2,m3), is called almost complete tripartite graph. A graph (K) over tilde (m1),(m2),(m3) that can be decomposed into two isomorphic factors with a given diameter d is called d-halvable. We prove that (K) over tilde (m1,m2,m3) is d-halvable for a finite d only if d less than or equal to 5 and completely determine all triples 2m ' (1) + 1, 2m ' (2) +1, 2m ' (3) for which there exist d-halvable almost complete tripartite graphs for diameters 3,4 and 5, respectively

Note on cyclic decompositions of complete bipartite graphs into cubes

So far, the smallest complete bipartite graph which was known to have a cyclic decomposition into cubes $Q_d$ of a given dimension d was $K_{d2^{d-1}, d2^{d-2}}$. We improve this result and show that also $K_{d2^{d-2}, d2^{d-2}}$ allows a cyclic decomposition into $Q_d$. We also present a cyclic factorization of $K_{8,8}$ into Q₄

Range of diameters of complementary factors of almost complete tripartite graphs

A complete tripartite graph without one edge, (K) over tilde(m1),(m2),(m3), is called almost complete tripartite graph. A graph (K) over tilde(m1),(m2),(m3) that can be decomposed into two isomorphic factors with a given diameter d is called d-halvable. We prove that (K) over tilde(m1,m2,m3) is d-halvable for a finite d only if d less than or equal to 5