The direct product of a star and a path is antimagic

Abstract

A graph $G$ is antimagic if there exists a bijection $f$ from $E(G)$ to $\left\{1,2, \dots,|E(G)|\right\}$ such that the vertex sums for all vertices of $G$ are distinct, where the vertex sum is defined as the sum of the labels of all incident edges. Hartsfield and Ringel conjectured that every connected graph other than $K_2$ admits an antimagic labeling. It is still a challenging problem to address antimagicness in the case of disconnected graphs. In this paper, we study antimagicness for the disconnected graph that is constructed as the direct product of a star and a path.

Figures (2)

• Figure 1: An antimagic labeling of of $K_{1,3}\times P_{8}$.
• Figure 2: An antimagic labeling of of $K_{1,3}\times P_{7}$.

Theorems & Definitions (16)

• Theorem 1
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof
• Theorem 7
• Lemma 8