(a,d)-Antimagic Ascending Subgraph Decomposition of Graphs
Sigit Pancahayani, Rinovia MG Simanjuntak

Institut Teknologi Bandung

Abstract

Let -G- be a finite graph of positive size -q- and let -n- be a positive integer with -{n+1 \choose 2} \leq q < {n+2 \choose 2}-. -G- is said to have an ascending subgraph decomposition (ASD) if -G- can be decomposed into -n- subgraphs -G_1, G_2, \ldots, G_n- without isolated vertices such that -G_i- is isomorphic to a proper subgraph of -G_{i+1}- for -1 \leq i \leq n-1-.\\
%Furthermore, -E(G_i)\cap E(G_j)=\emptyset- for -i\neq j- and -\cup_{i=1}^{n}G_i=G-.

In this presentation, we introduce a new type of antimagic labeling as follow. Let -G- admits an ASD. If -f- is a bijection from -V(G)\cup E(G)- into -\{1,2,\ldots,|V(G)|+|E(G)|\}- such that for every -G_i- the weight -w_i=\sum_{v\in V(G_i)}f(v)+\sum_{e\in E(G_i)}f(e)- forms an arithmetic progression with first term -a- and common difference -d-, then -f- is called an -(a,d)--antimagic-ASD labeling. In such a case,
-G- is called -(a,d)--antimagic-ASD.\\

Furthermore, we investigate some classes of -(a,d)--antimagic-ASD graphs.

Keywords: ascending subgraph decomposition (ASD), antimagic labeling, -(a,d)--antimagic-ASD labeling

Topic: MATHEMATICS AND STATISTICS