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13
A dynamic survey of graph labellings
 Electron. J. Combin., Dynamic Surveys(6):95pp
, 2001
"... A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done ..."
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A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.
Antimagic graphs via the combinatorial nullstellensatz
 J. Graph Theory
, 2005
"... An antimagic labelling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1,..., m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it has an anti ..."
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An antimagic labelling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1,..., m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it has an antimagic labelling. In [10], Ringel conjectured that every simple connected graph, other than K2, is antimagic. We prove several special cases and variants of this conjecture. Our main tool is the Combinatorial NullStellenSatz (c.f. [1]). 1
Regular graphs of odd degree are antimagic
, 2013
"... An antimagic labeling of a graph G with m edges is a bijection from E(G) to {1, 2,...,m} such that for all vertices u and v, the sum of labels on edges incident to u differs from that for edges incident to v. Hartsfield and Ringel conjectured that every connected graph other than the single edge K2 ..."
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An antimagic labeling of a graph G with m edges is a bijection from E(G) to {1, 2,...,m} such that for all vertices u and v, the sum of labels on edges incident to u differs from that for edges incident to v. Hartsfield and Ringel conjectured that every connected graph other than the single edge K2 has an antimagic labeling. We prove this conjecture for regular graphs of odd degree. 1
A New Class of Antimagic Cartesian Product Graphs ∗
"... An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,..., m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called ..."
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An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,..., m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [5] conjectured that every simple connected graph, but K2, is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In particular, by combining this result and the antimagicness result on toroidal grids (Cartesian products of two cycles) in [7], all Cartesian products of two or more regular graphs of positive degree can be proved to be antimagic.
Lattice grids and prisms are antimagic
 Theoretical Computer Science
"... An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,...,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called ..."
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An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,...,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every connected graph, but K2, is antimagic. In 2004, N. Alon et al showed that this conjecture is true for nvertex graphs with minimum degree Ω(log n). They also proved that complete partite graphs (other than K2) and nvertex graphs with maximum degree at least n − 2 are antimagic. Recently, Wang showed that the toroidal grids (the Cartesian products of two or more cycles) are antimagic. Two open problems left in Wang’s paper are about the antimagicness of lattice grid graphs and prism graphs, which are the Cartesian products of two paths, and of a cycle and a path, respectively. In this article, we prove that these two classes of graphs are antimagic, by constructing such antimagic labelings.
Coloring and Labeling Problems on Graphs
, 2007
"... This thesis studies both several extremal problems about coloring of graphs and a labeling problem on graphs. We consider colorings of graphs that are either embeddable in the plane or have low maximum degree. We consider three problems: coloring the vertices of a graph so that no adjacent vertices ..."
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This thesis studies both several extremal problems about coloring of graphs and a labeling problem on graphs. We consider colorings of graphs that are either embeddable in the plane or have low maximum degree. We consider three problems: coloring the vertices of a graph so that no adjacent vertices receive the same color, coloring the edges of a graph so that no adjacent edges receive the same color, and coloring the edges of a graph so that neither adjacent edges nor edges at distance one receive the same color. We use the model where colors on vertices must be chosen from assigned lists and consider the minimum size of lists needed to guarantee the existence of a proper coloring. More precisely, a list assignment function L assigns to each vertex a list of colors. A proper Lcoloring is a proper coloring such that each vertex receives a color from its list. A graph is klistcolorable if it has an Lcoloring for every list assignment L that assigns each vertex a list of size k. The list chromatic number χl(G) of a graph G is the minimum k such that G is klistcolorable. We also call the list chromatic number the choice number of the graph. If a graph is klistcolorable, we call it kchoosable. The elements of a graph are its vertices and edges. A proper total coloring of a graph is a coloring
Antimagic labelings of regular bipartite graphs: An application of the Marriage Theorem
, 2007
"... A labeling of a graph is a bijection from E(G) to the set {1,2,..., E(G)}. A labeling is antimagic if for any distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it has an antimagic l ..."
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A labeling of a graph is a bijection from E(G) to the set {1,2,..., E(G)}. A labeling is antimagic if for any distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than K2 is antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem. 1
Products of Regular Graphs are Antimagic
, 2006
"... An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,...,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called ..."
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An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,...,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [5] conjectured that every simple connected graph, but K2, is antimagic. We prove that the Cartesian product graphs G1 ×G2 (where G1 is a connected k1regular graph and G2 is a graph with maximum degree at most k2, minimum degree at least one) are antimagic, provided that k1 is odd and k2 1−k1 2 ≥ k2, or, k1 is even and k2 1 2 ≥ k2 and k1, k2 are not both equal to 2. By combining the above result and the antimagicness result on toroidal grids (Cartesian products of two cycles) in [9], we obtain that all Cartesian products of two or more regular graphs are antimagic. We also give a generalization of the above antimagicness result on G1 ×G2, for which G1 is not necessarily connected.
Cartesian Products of Regular Graphs are Antimagic
, 2006
"... An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,...,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called ..."
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An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,...,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [4] conjectured that every simple connected graph, but K2, is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In addition, by combining this result and the antimagicness result on toroidal grids (Cartesian products of two cycles) in [6], all Cartesian products of two or more regular graphs can be proved to be antimagic.