6 research outputs found
Approximate results for rainbow labelings
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Approximate results for rainbow labelings
The final publication is available at Springer via https://doi.org/10.1007/s10998-016-0151-2]A simple graph G=(V,E) is said to be antimagic if there exists a bijection f:E¿[1,|E|] such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection f:V¿[1,|V|], such that ¿x,y¿V, ¿xi¿N(x)f(xi)¿¿xj¿N(y)f(xj). Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval [1,2n+m-4] and, for trees with k inner vertices, in the interval [1,m+k]. In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree ¿ in the interval [1,n+t(n-t)], where t=min{¿,¿n/2¿}, and, for trees with k leaves, in the interval [1,3n-4k]. In particular, all trees with n=2k vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.Peer ReviewedPostprint (author's final draft
Antimagic Labelings of Caterpillars
A -antimagic labeling of a graph is an injection from to
such that all vertex sums are pairwise distinct, where
the vertex sum at vertex is the sum of the labels assigned to edges
incident to . We call a graph -antimagic when it has a -antimagic
labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel
conjectured that every simple connected graph other than is antimagic,
but the conjecture is still open even for trees. Here we study -antimagic
labelings of caterpillars, which are defined as trees the removal of whose
leaves produces a path, called its spine. As a general result, we use
constructive techniques to prove that any caterpillar of order is -antimagic. Furthermore, if is a caterpillar with a
spine of order , we prove that when has at least leaves or consecutive vertices of degree at
most 2 at one end of a longest path, then is antimagic. As a consequence of
a result by Wong and Zhu, we also prove that if is a prime number, any
caterpillar with a spine of order , or is -antimagic.Comment: 13 pages, 4 figure
Approximate results for rainbow labelings
Article de recerc