Antimagic labelings on graphs with ascending subgraph decomposition

Let t and q be positive integers that satisfy \binom{t+۱}{۲} \leq q< \binom{t+۲}{۲} and G be a simple and finite graph of size q. G is said to be an ascending subgraph decomposition (ASD) graph if G can be decomposed into t subgraphs H_۱, H_۲,\ldots,H_t without isolated vertices such that H_i is isomorphic to a proper subgraph of H_{i+۱}, for ۱ \leq i \leq t-۱. In this paper, we introduce a new type of antimagic labeling based on the notion of ASD. Let G be an ASD graph and f:V(G)\cup E(G) \rightarrow \{۱,۲,\ldots,\lvert V(G)\rvert+\lvert E(G)\rvert\} a bijection. The weight of a subgraph H_i (۱\leq i\leq t) is w(H_i)=\sum_{v\in V(H_i)}f(v)+\sum_{e\in E(H_i)}f(e). If the weights of all H_is (۱\leq i\leq t) form an arithmetic progression with the smallest weight a and common difference d, then f is called an (a,d)-ASD antimagic labeling and G is an (a,d)-ASD antimagic graph. We provide an upper bound for d in an (a,d)-ASD antimagic graph. We define and utilize the (t,\delta)-ascending antibalanced multisets to label some product graphs, including disjoint union, vertex amalgamation, edge amalgamation, subgraph amalgamation, and extended chain of graphs.

کلیدواژه ها:

ascending subgraph decomposition (ASD) ، antimagic labeling ، (a ، d)-ASD antimagic labeling