Oberwolfach rectangular table negotiation problem

AbstractWe completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H(k,3) be a bipartite graph with bipartition X={x1,x2,…,xk},Y={y1,y2,…,yk} and edges x1y1,x1y2,xkyk−1,xkyk, and xiyi−1,xiyi,xiyi+1 for i=2,3,…,k−1. We completely characterize all complete bipartite graphs Kn,n that can be factorized into factors isomorphic to G=mH(k,3), where k is odd and mH(k,3) is the graph consisting of m disjoint copies of H(k,3)

Regular handicap tournaments of high degree

A handicap distance antimagic labeling of a graph $G=(V,E)$ with $n$ vertices is a bijection ${f}: V\to \{ 1,2,\ldots ,n\}$ with the property that ${f}(x_i)=i$ and the sequence of the weights $w(x_1),w(x_2),\ldots,w(x_n)$ (where $w(x_i)=\sum\limits_{x_j\in N(x_i)}f(x_j)$) forms an increasing arithmetic progression with difference one. A graph $G$ is a {\em handicap distance antimagic graph} if it allows a handicap distance antimagic labeling. We construct $(n-7)$-regular handicap distance antimagic graphs for every order $n\equiv2\pmod4$ with a few small exceptions. This result complements results by Kov\'a\v{r}, Kov\'a\v{r}ov\'a, and Krajc~[P. Kov\'a\v{r}, T. Kov\'a\v{r}ov\'a, B. Krajc, On handicap labeling of regular graphs, manuscript, personal communication, 2016] who found such graphs with regularities smaller than $n-7$

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

A $d$-handicap distance antimagic labeling of a graph $G=(V,E)$ with $n$ vertices is a bijection $f:V\to \{1,2,\ldots ,n\}$ with the property that $f(x_i)=i$ and the sequence of weights $w(x_1),w(x_2),\ldots,w(x_n)$ (where $w(x_i)=\sum_{x_i x_j\in E}f(x_j)$) 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 $n\equiv8\pmod{16}$, $n\geq56$ and $6\leq k\leq n-50$

Distance Magic Graphs - a Survey

Let <i>G = (V;E)</i> be a graph of order n. A bijection <i>f : V → {1, 2,...,n} </i>is called <i>a distance magic labeling </i>of G if there exists a positive integer k such that <i>Σ f(u) = k </i> for all <i>v ε V</i>, where <i>N(v)</i> is the open neighborhood of v. The constant k is called the magic constant of the labeling f. Any graph which admits <i>a distance magic labeling </i>is called a distance magic graph. In this paper we present a survey of existing results on distance magic graphs along with our recent results,open problems and conjectures.DOI : http://dx.doi.org/10.22342/jims.0.0.15.11-2

Z2nm-supermagic labeling of Cn#Cm

A Γ-supermagic labeling of a graph G = (V, E) with ∣E∣ = k is a bijection from E to an Abelian group Γ of order k such that the sum of labels of all incident edges of every vertex x ∈ V is equal to the same element μ ∈ Γ. We present a Z2nm-supermagic labeling of Cartesian product of two cycles, Cn□Cm for n odd. This along with an earlier result by Ivančo proves that a Z2nm-supermagic labeling of Cn□Cm exists for every n, m ≥ 3.</p

Alpha labelings of full hexagonal caterpillars

Barrientos and Minion (2015) introduced the notion of generalized snake polyomino graphs and proved that when the cells are either squares or hexagons, then they admit an alpha labeling. Froncek et al. (2014) generalized the notion by introducing straight simple polyominal caterpillars with square cells and proved that they also admit an alpha labeling. We introduce a similar family of graphs called full hexagonal caterpillars and prove that they also admit an alpha labeling. This implies that every full hexagonal caterpillar with n edges decomposes the complete graph K2kn+1 for any positive integer k

Decomposition of Certain Complete Bipartite Graphs into Prisms

Häggkvist [6] 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 [1] Cichacz and Froncek established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph Kn,n into generalized prisms of order 2n. In [2] and [3] Cichacz, Froncek, and Kovar showed decompositions of K3n/2,3n/2 into generalized prisms of order 2n. In this paper we prove that K6n/5,6n/5 is decomposable into prisms of order 2n when n ≡ 0 (mod 50)