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A note on incomplete regular tournaments with handicap two of order n ≡8 (mod 16)

Froncek D.

Opuscula Mathematica

2017

Vol. 37, no. 4

557--566

A d-handicap distance antimagic labeling of a graph G = (V, E) with n vertices is a bijection ƒ : V → 1, 2,..., n } with the property that ƒ(xi) = i and the sequence of weights w(x1), w(x2),... , w(xn) (where [formula] ) forms an increasing arithmetic progression with common difference d. A graph G is a d-handicap distance antimagic graph if it allows a d-handicap distance antimagic labeling. We construct a class of k-regular 2-handicap distance antimagic graphs for every order n ≡8 (mod 16), n ≥56 and 6 ≥ k ≥ n — 50.

Decomposition of complete graphs into small graphs

Froncek D.

Opuscula Mathematica

2010

Vol. 30, no. 3

277-280

In 1967, A. Rosa proved that if a bipartite graph G with n edges has an α-labeling, then for any positive integer p the complete graph K(2np+1) can be cyclically decomposed into copies of G. This has become a part of graph theory folklore since then. In this note we prove a generalization of this result. We show that every bipartite graph H which decomposes K(k) and K(m) also decomposes K(km).

α2-labeling of graphs

Froncek D.

Opuscula Mathematica

2009

Vol. 29, no. 4

393-397

We show that if a graph G on n edges allows certain special type of rosy labeling (a.k.a. rho;-labeling), called α2-labeling, then for any positive integer k the complete graph K2nk+1 can be decomposed into copies of G. This notion generalizes the α-labeling introduced in 1967 by A. Rosa.

3-halvable almost complete tripartite graphs

Froncek D.

Opuscula Mathematica

2001

Vol. 21

43-57

A complete tripartite graph without one edge, Km1,m2,m3, is called almost complete tripartite graph. A graph Km1,m2,m3 that can be decomposed into two isomorphic factors with a given diameter d is called d-halvable. We completely determine all triples 2m'1 +1, 2m'2 +1, 2m'3, for which there exists a 3-halvable almost complete tripartite graph.

Graf trójdzielny zupełny bez jednego wierzchołka, Km1,m2,m3, jest nazywany prawie zupełnym grafem trójdzielnym. Graf Km1,m2,m3, który można rozłożyć na dwa czynniki izomorficzne mające zadaną średnicę d jest nazywany dającym się d-połowić. Wyznaczono wszystkie trójki 2m'1 + 1, 2m'2 + 1, 2m'3, dla których istnieje prawie zupełny graf trójdzielny dający się 3-połowić.