Speaker: Professor Po-Yi Huang (National Cheng Kung University)
Title: Weighted-1-antimagic graphs of prime power order
Time: 2010-07-15 (Thu.)  15:00 - 16:00
Place: Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)
Abstract: We say $G$ is weighted-$k$-antimagic if for any vertex weight function $w: V \to \mathbb{N}$, there is an injection $f: E \to \{1,2,\ldots, |E|+k\}$ such that for any two distinct vertices $u$ and $v$, $\sum_{e \in E(v)}f(e)+w(v) \ne \sum_{e \in E(u)}f(e)+w(u)$. There are connected graphs $G \ne K_2$ which are not weighted-$1$-antimagic. It was asked whether every connected graph other than $K_2$ is weighted-$2$-antimagic, and whether every connected graph on an odd number of vertices is weighted-$1$-antimagic. By restricting to graphs of prime prime order, we improve this result in two directions: if $G$ has odd prime power order $p^z$ and has total domination number $2$ with the degree of one vertex in the total dominating set not a multiple of $p$, then $G$ is weighted-$1$-antimagic. If $G$ has odd prime ($\neq 3$) power order and has maximum degree at least $|V(G)|-3$, then $G$ is weighted-$1$-antimagic.
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