Antimagic labelling of vertex weighted graphs

Abstract

Suppose G is a graph, k is a non-negative integer. We say G is k-antimagic if there is an injection f : E →{1, 2, . . . , |E|+k} such that for any two distinct vertices u and v, e∈E(v) f (e) = e∈E (u) f (e). We say G is weighted-k-antimagic if for any vertex weight function w : V → N, there is an injection f : E →{1, 2, . . . , |E|+k} such that for any two distinct vertices u and v, e∈E(v) f (e)+w(v) = e∈E(u) f (e)+w(u). A well-known conjecture asserts that every connected graph G = K 2 is 0-antimagic. On the other hand, there are connected graphs G = K 2 which are not weighted-1-antimagic. It is unknown whether every connected graph G = K 2 is weighted-2-antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted-2-antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted-1-antimagic.

Antimagic labelings of caterpillars

“…Furthermore, if C is a caterpillar with a spine of order s, we prove that when C has at least (3s + 1)/2 leaves or (s − 1)/2 consecutive vertices of degree at most 2 at one end of a longest path, then C is antimagic. As a consequence of a result by Wong and Zhu [22], we also prove that if p is a prime number, any caterpillar with a spine of order p, p − 1 or p − 2 is 1-antimagic. *…”

Neighbor Distinguishing Edge Colorings via the Combinatorial Nullstellensatz

“…, v k−1 , provided by our initial coloring of the graph G−v. Accidentally, such polynomial has already been investigated in the context of weighted antimagic labelings of stars by Wong and Zhu (see [12] for details), whose approach also inspired our reasoning. Note that the polynomial P has degree (k − 1) 2 .…”

The adjacent vertex distinguishing total coloring of planar graphs without adjacent 4-cycles

“…The family of such problems includes e.g. vertex-coloring [k]-edge weightings (Kalkowski et al 2010), neighbor sum distinguishing edge (total) coloring (Dong et al 2014;Qu et al 2015;Wang and Yan 2014;, total weight choosability (Przybyło and Woźniak 2011;Wong and Zhu 2011), magic and antimagic labellings Wong and Zhu 2012), the irregularity strength (Przybyło 2008;Seamone 2012) and additive colorings (Bartnicki et al 2014).…”