On some third parts of nearly complete digraphs

AbstractFor the complete digraph DKn with n⩾3, its half as well as its third (or near-third) part, both non-self-converse, are exhibited. A backtracking method for generating a tth part of a digraph is sketched. It is proved that some self-converse digraphs are not among the near-third parts of the complete digraph DK5 in four of the six possible cases. For n=9+6k,k=0,1,…, a third part D of DKn is found such that D is a self-converse oriented graph and all D-decompositions of DKn have trivial automorphism group

Kite systems of order 8;Embedding of kite systems into bowtie systems

This article consist of two parts. In the first part, we enumerate the kite systems of order 8; in the second part, we consider embedding kite systems into bowtie systems

Solution to the problem of Kubesa

An infinite family of T-factorizations of complete graphs $K_{2n}$, where 2n = 56k and k is a positive integer, in which the set of vertices of T can be split into two subsets of the same cardinality such that degree sums of vertices in both subsets are not equal, is presented. The existence of such T-factorizations provides a negative answer to the problem posed by Kubesa

CIRCULANTS AND THE CHROMATIC INDEX OF STEINER TRIPLE SYSTEMS

Abstract. We complete the determination of the chromatic number of 6valent circulants of the form C(n; a, b, a+b) and show how this can be applied to improving the upper bound on the chromatic index of cyclic Steiner triple systems

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

Decompositions of nearly complete digraphs into t isomorphic parts

An arc decomposition of the complete digraph Kₙ into t isomorphic subdigraphs is generalized to the case where the numerical divisibility condition is not satisfied. Two sets of nearly tth parts are constructively proved to be nonempty. These are the floor tth class ( Kₙ-R)/t and the ceiling tth class ( Kₙ+S)/t, where R and S comprise (possibly copies of) arcs whose number is the smallest possible. The existence of cyclically 1-generated decompositions of Kₙ into cycles $^{→}C_{n-1}$ and into paths $^{→}Pₙ$ is characterized

Decompositions of a complete multidigraph into almost arbitrary paths

For n ≥ 4, the complete n-vertex multidigraph with arc multiplicity λ is proved to have a decomposition into directed paths of arbitrarily prescribed lengths ≤ n - 1 and different from n - 2, unless n = 5, λ = 1, and all lengths are to be n - 1 = 4. For λ = 1, a more general decomposition exists; namely, up to five paths of length n - 2 can also be prescribed