On sequential labelings of graphs

A valuation on a simple graph G is an assignment of labels to the vertices of G which induces an assignment of labels to the edges of G. β-valuations, also called graceful labelings, and α-valuations, a subclass of graceful labelings, have an extensive literature; harmonious labelings have been introduced recently by Graham and Sloane. This paper introduces sequential labelings, a subclass of harmonious labelings, and shows that any tree admitting an α-valuation also admits a sequential labeling and hence is harmonious. Constructions are given for new families of graceful and sequential graphs, generalizing some earlier results. Finally, a conjecture of Frucht is shown to be wrong by exhibiting several graceful labelings of wheels in which the center label is larger than previously thought possible.     

On Rosa-type labelings and cyclic graph decompositions

A labeling (or valuation) of a graph G is an assignment of integers to the vertices of G subject to certain conditions. A hierarchy of graph labelings was introduced by Rosa in the late 1960s. Rosa showed that certain basic labelings of a graph G with n edges yielded cyclic G-decompositions of K _2n+1 while other stricter labelings yielded cyclic G-decompositions of K _2nx+1 for all natural numbers x. Rosa-type labelings are labelings with applications to cyclic graph decompositions. We survey various Rosa-type labelings and summarize some of the related results. (Communicated by Peter Horák)     

Labeling constructions using digraph products

In this paper we study the edge-magicness of graphs with equal size and order, and we use such graphs and digraph products in order to construct labelings of different classes and of different graphs. We also study super edge-magic labelings of 2$2$-regular graphs with exactly two components and their implications to other labelings.     

Simply sequentially additive labelings of 2-regular graphs

We conjecture that any 2-regular simple graph has an SSA labeling. We provide several special cases to support our conjecture. Most of our constructions are based on Skolem sequences and on an extension of it. We establish a connection between simply sequentially additive labelings of 2-regular graphs and ordered graceful labelings of spiders.     

图P_n~2的-优美性

本文针对［１］中提出的猜测“我们猜测Ｐｎ２是优美的,尽管标号看来更复杂”,对Ｐｎ２的标号作了一些工作,证明了Ｐｎ２：是－优美的．     

图Cn及其r-冠的新的优美标号

研究了关于图的r-冠的优美标号的一个问题,证明了：当n≡0,3（mod 4）时,图Cn及其r-冠是优美图,所给出的新的优美标号不同于现有文献中得到的结果.进而证明了当n≡0（mod 4）时,图Cn及其r-冠也是交错图.     

Geodesic Embeddings and Planar Graphs

Schnyder labelings are known to have close links to order dimension and drawings of planar graphs. It was observed by Ezra Miller that geodesic embeddings of planar graphs are another class of combinatorial or geometric objects closely linked to Schnyder labelings. We aim to contribute to a better understanding of the connections between these objects. In this article we prove • a characterization of 3-connected planar graphs as those graphs admitting rigid geodesic embeddings, • a bijection between Schnyder labelings and rigid geodesic embeddings, • a strong version of the Brightwell–Trotter theorem. This revised version was published online in June 2006 with corrections to the Cover Date.     

Group labelings of graphs

Given a graph Γ an abelian group G, and a labeling of the vertices of Γ with elements of G, necessary and sufficient conditions are stated for the existence of a labeling of the edges in which the label of each vertex equals the product of the labels of its incident edges. Such an edge labeling is called compatible. For vertex labelings satisfying these conditions, the set of compatible edge labelings is enumerated.     

Graph puzzles,homotopy, and the alternating group

The so-called 15-puzzle may be generalized to a puzzle based on an arbitrary graph. We consider labelings or colorings of the vertices and the operation of switching one distinguished label with a label on an adjacent vertex. Starting from a given labeling, iterations of this operation allow one to obtain all, or exactly half, of the labelings on a non-separable graph (with the polygons and one other graph as exceptions).     

On Some Extremal Properties of Finite Ultrametric Spaces

We characterize finite ultrametric spaces having the strictly n-ary representing trees and finite ultrametric spaces having the representing trees with injective internal labelings by their extremal properties.     

Crystal Graphs of Higher Level <Emphasis Type="Italic">q</Emphasis>-deformed Fock Spaces,Lusztig <Emphasis Type="Italic">a</Emphasis>-values and Ariki–Koike Algebras

We show that the different labelings of the crystal graph for irreducible highest weight -modules lead to different labelings of the simple modules for non semisimple Ariki–Koike algebras by using Lusztig a-values. Presented by Peter Littelman.     

Vertex-magic labelings of regular graphs II

Previously the first author has shown how to construct vertex-magic total labelings (VMTLs) for large families of regular graphs. The construction proceeds by successively adding arbitrary 2-factors to a regular graph of order n which possesses a strong VMTL, to produce a regular graph of the same order but larger size. In this paper, we exploit this construction method. We are able to show that for any r≥4, every r-regular graph of odd order n≤17 has a strong VMTL. We show how to produce strong labelings for some families of 2-regular graphs since these are used as the starting points of our construction. While even-order regular graphs are much harder to deal with, we introduce ‘mirror’ labelings which provide a suitable starting point from which the construction can proceed. We are able to show that several large classes of r-regular graphs of even order (including some Hamiltonian graphs) have VMTLs.     

关于图P_n~3优美性的研究

在n个顶点的路Pn上,当且仅当两点的距离为3时增加一条边,所得的图称为P3n,本文给出了图P3n(n≥4)的优美标号,从而证明了P3n都是优美图.     

Labelings of Lee and Hamming spaces

The labeling of the Hamming Space by the rotation group and its coordinate-wise extension to give rise to the concept of -linearity. Attempts to extend this concept have been done in different ways. We deal with a natural extension question: Is there any pattern of a cyclic group G labeling of with the Hamming or Lee metric? The answer is no. Actually, we show here that Lee spaces do not allow even labelings by abelian groups, what lead us to construct labelings by semi-direct products of abelian groups. Labelings of general Hamming spaces and of Reed–Muller codes RM(1,m) are characterized here in the context of isometry groups.     

On Cyclic Decompositions of Complete Graphs into Tripartite Graphs

We introduce two new labelings for tripartite graphs and show that if a graph G with n edges admits either of these labelings, then there exists a cyclic G‐decomposition of for every positive integer x. We also show that if G is the union of two vertext‐disjoint cycles of odd length, other than , then G admits one of these labelings.     

A survey: Recent results,conjectures, and open problems in labeling graphs

In this paper we organize and summarize much of the work done on graceful and harmonious labelings of graphs. Many open problems and conjectures are included.     

Graphs with flexible labelings allowing injective realizations

We consider realizations of a graph in the plane such that the distances between adjacent vertices satisfy the constraints given by an edge labeling. If there are infinitely many such realizations, counted modulo rigid motions, the labeling is called flexible. The existence of a flexible labeling, possibly non-generic, has been characterized combinatorially by the existence of a so called NAC-coloring. Nevertheless, the corresponding realizations are often non-injective. In this paper, we focus on flexible labelings with infinitely many injective realizations. We provide a necessary combinatorial condition on existence of such a labeling based also on NAC-colorings of the graph. By introducing new tools for the construction of such labelings, we show that the necessary condition is also sufficient up to 8 vertices, but this is not true in general for more vertices.     

A characterization of hypercacti

In [M. Sonntag, Antimagic vertex labelings of hypergraphs, Discrete Math. 247 (2002) 187-199] the graph theoretic notion of a cactus is generalized to hypergraphs. We present some elementary properties and give a characterization of hypercacti.     

Strong labelings of linear forests

A （p, q）-graph G is called super edge-magic if there exists a bijective function f ： V（G） U E（G） →{1, 2 p＋q} such that f（u）＋ f（v）＋f（uv） is a constant for each uv C E（G） and f（Y（G）） = {1,2,...,p}. In this paper, we introduce the concept of strong super edge-magic labeling as a particular class of super edge-magic labelings and we use such labelings in order to show that the number of super edge-magic labelings of an odd union of path-like trees （mT）, all of them of the same order, grows at least exponentially with m.