TY - JOUR

T1 - An application of the combinatorial Nullstellensatz to a graph labelling problem

AU - Hefetz, Dan

AU - Saluz, Annina

AU - Tran, Huong T.T.

PY - 2010/9

Y1 - 2010/9

N2 - An antimagic labelling of a graph G with m edges and n vertices is a bijection from the set of edges of G to the set of integers {1,...,m}, such that all nvertex 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 admits an antimagic labelling. In N. Hartsfield and G. Ringle, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, Ringel has conjectured that every simple connected graph, other than K2, is antimagic. In this article, we prove a special case of this conjecture. Namely, we prove that if G is a graph on n = pk vertices, where p is an odd prime and k is a positive integer that admits a Cp-factor, then it is antimagic. The case p=3 was proved in D. Hefetz, J Graph Theory 50(2005), 263-272. Our main tool is the combinatorial Nullstellensatz [N. Alon, Combin Probab Comput 8(1-2) (1999), 7-29].

AB - An antimagic labelling of a graph G with m edges and n vertices is a bijection from the set of edges of G to the set of integers {1,...,m}, such that all nvertex 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 admits an antimagic labelling. In N. Hartsfield and G. Ringle, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, Ringel has conjectured that every simple connected graph, other than K2, is antimagic. In this article, we prove a special case of this conjecture. Namely, we prove that if G is a graph on n = pk vertices, where p is an odd prime and k is a positive integer that admits a Cp-factor, then it is antimagic. The case p=3 was proved in D. Hefetz, J Graph Theory 50(2005), 263-272. Our main tool is the combinatorial Nullstellensatz [N. Alon, Combin Probab Comput 8(1-2) (1999), 7-29].

KW - Combinatorial Nullstellensatz

KW - Graph labelling

UR - http://www.scopus.com/inward/record.url?scp=77957054898&partnerID=8YFLogxK

U2 - 10.1002/jgt.20466

DO - 10.1002/jgt.20466

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AN - SCOPUS:77957054898

SN - 0364-9024

VL - 65

SP - 70

EP - 82

JO - Journal of Graph Theory

JF - Journal of Graph Theory

IS - 1

ER -