2,537 research outputs found
On the number of regular edge labelings
We prove that any irreducible triangulation on n vertices has O (4:6807n ) regular edge labeling,s and that there are irreducible triangulations on n vertices with (3:0426n ) regular edge labelings. Our upper bound relies on a novel application of Shearer's entropy lemma. As an example of the wider applicability of this technique, we also improve the upper bound on the number of 2-orientations of a quadrangulation to O (1:87n ). Keywords: Counting; Regular edge labeling; Shearer's entropy lemm
On Regular Graphs Optimally Labeled with a Condition at Distance Two
For positive integers , the -number of graph Gis the smallest span among all integer labelings of V(G) such that vertices at distance two receive labels which differ by at least k and adjacent vertices receive labels which differ by at least j. We prove that the -number of any r-regular graph is no less than the -number of the infinite r-regular tree . Defining an r-regular graph G to be -optimal if and only if , we establish the equivalence between -optimal graphs and r-regular bipartite graphs with a certain edge coloring property for the case . The structure of -regular optimal graphs for is investigated, with special attention to . For the latter, we establish that a (2,1,r)-optimal graph, through a series of edge transformations, has a canonical form. Finally, we apply our results on optimality to the derivation of the -numbers of prisms
Perfect (super) Edge-Magic Crowns
A graph G is called edge-magic if there is a bijective function f from the set of vertices and edges to the set {1,2,…,|V(G)|+|E(G)|} such that the sum f(x)+f(xy)+f(y) for any xy in E(G) is constant. Such a function is called an edge-magic labelling of G and the constant is called the valence. An edge-magic labelling with the extra property that f(V(G))={1,2,…,|V(G)|} is called super edge-magic. A graph is called perfect (super) edge-magic if all theoretical (super) edge-magic valences are possible. In this paper we continue the study of the valences for (super) edge-magic labelings of crowns Cm¿K¯¯¯¯¯n and we prove that the crowns are perfect (super) edge-magic when m=pq where p and q are different odd primes. We also provide a lower bound for the number of different valences of Cm¿K¯¯¯¯¯n, in terms of the prime factors of m.Postprint (updated version
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