a graceful labeling of a graph g of size n is an injective assignment of integers from {0, 1,..., n} to the vertices of g, such that when each edge of g has assigned a weight, given by the absolute dierence of the labels of its end vertices, the set of weights is {1, 2,..., n}. if a graceful labeling f of a bipartite graph g assigns the smaller labels to one of the two stable sets of g, then f...

For a graph G = (V,E) we consider vertex-k-labellings f : V → {1, 2, . . . , k} for which the induced edge weighting w : E → {2, 3, . . . , 2k} with w(uv) = f(u) + f(v) is injective or surjective or both. We study the relation between these labellings and the number theoretic notions of an additive basis and a Sidon set, present a new construction for a so-called restricted additive basis and d...

For a given simple graph an average labelling is de)ned. The graphs with average labellings and all the admissible average labellings for such graphs are characterized. c © 2001 Elsevier Science B.V. All rights reserved.

In this work, we consider equitable proper labellings of graphs, which were recently introduced by Baudon, Pilśniak, Przybyło, Senhaji, Sopena, and Woźniak. Given a graph G, the goal is to assign labels edges so that (1) no two adjacent vertices are incident same sum labels, (2) every assigned about number times. Particularly, aim at designing such k-labellings G with k being as small possible....

‎a $(p‎,‎q)$ graph $g$ is said to have a $k$-odd mean‎ ‎labeling $(k ge 1)$ if there exists an injection $f‎ : ‎v‎ ‎to {0‎, ‎1‎, ‎2‎, ‎ldots‎, ‎2k‎ + ‎2q‎ - ‎3}$ such that the‎ ‎induced map $f^*$ defined on $e$ by $f^*(uv) =‎ ‎leftlceil frac{f(u)+f(v)}{2}rightrceil$ is a‎ ‎bijection from $e$ to ${2k - ‎‎‎1‎, ‎2k‎ + ‎1‎, ‎2k‎ + ‎3‎, ‎ldots‎, ‎2‎ ‎k‎ + ‎2q‎ - ‎3}$‎. ‎a graph that admits $k$...

Here we denote a diameter six tree by (c; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr), where c is the center of the tree; ai, i = 1, 2, . . . ,m, bj, j = 1, 2, . . . , n, and ck, k = 1, 2, . . . , r are the vertices of the tree adjacent to c; each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Here we give graceful lab...

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We provide binary labellings for the angles of quadrangulations and plane Laman graphs which are in analogy with Schnyder labellings for triangulations [W. Schnyder, Proc. 1st ACM-SIAM Symposium on Discrete Algorithms, 1990].

The problem of radio channel assignments with multiple levels of interference can be modelled using graph theory. The theory of integer vertex-labellings of graphs with distance conditions has been investigated for several years now, and the authors recently introduced a new model of real number labellings that is giving deeper insight into the problems. Here we present an overview of the recen...

Ryan Jones, Western Michigan University We introduce a modular edge-graceful labeling of a graph a dual concept to the common graceful labeling. A 1991 conjecture known as the Modular Edge-Graceful Tree Conjecture states that every tree of order n where n 6≡ 2 (mod 4) is modular edge-graceful. We show that this conjecture is true. More general results and questions on this topic are presented.