The local antimagic (total) chromatic numbers of firecracker graphs and edge-corona product graphs

TL;DR

The paper determines the local antimagic (and total) chromatic numbers for firecracker graphs $F_{n,k}$ and for edge-corona products $G\diamond H$ with $G\in\{S_k,S_{k_1,k_2}\}$ and $H\in\{\overline{K_r},K_2\}$. It proves the exact value $\chi_{la}(F_{n,k})=nk-n+1$ for all $n,k\ge 2$, employing parity-based labelings and the labeling-matrix approach, while providing tight bounds for the local antimagic total chromatic number $\chi_{lat}(F_{n,k})$ (generally $2\le\chi_{lat}\le 3$, with many cases attaining 2). For edge-corona products, the results depend on the base graph and the attached graph: $\chi_{la}(S_k\diamond\overline{K_r})=3$, $3\le\chi_{la}(S_{k_1,k_2}\diamond\overline{K_r})\le 4$, $\chi_{la}(S_k\diamond rK_2)=4$, and $4\le\chi_{la}(S_{k_1,k_2}\diamond rK_2)\le 5$, with constructions provided via labeling matrices. These findings extend understanding of local antimagic colorings under important graph operations and offer systematic labeling-based proofs. The methods may guide future work on broader graph families and operations.

Abstract

Let G=(V(G),E(G)) be a connected simple graph with n vertices and m edges. A bijection f from the edge set of G to [m] is called a local antimagic labeling of G, if for any two adjacent vertices u and v in G, the sums of the weights of the edges associated with u and v ,respectively, are different. Similarly, A bijection g from the union of edge set and vertex set of G to [n+m] is called a local antimagic total labeling of G, if for any two adjacent vertices u and v in G, The sum of the weight of u and the weights of its incident edges differs from that of v. Obviously, any local antimagic (total) labeling induces a proper vertex-coloring of G when every vertex v is assigned the color w(v)(w_t(v)). The local antimagic (total) chromatic number of G, denoted by X_la(G)(X_lat(G)) , is defined as the minimum number of colors taken over all colorings induced by local antimagic (total) labelings of G. In this paper, we present the local antimagic (total) chromatic number of firecracker graph F_n,k, obtained by the concatenation of n k-stars by linking one leaf from each. Then we give the local antimagic chromatic number of the edge-corona product of two graphs G and H, where the graph is constructed by taking one copy of G and |E(G)| disjoint copies of H one-to-one assigned to each edge of G, and for every edge uv of G, joining u and v to every vertex of the copy of H associated to uv. For the graph studied here, G is a star S_k or a double star S_k1,k2, and H is an empty graph with r vertices or a complete graph K_2.

Figures (5)

• Figure 1: The firecracker graph $F_{5,4}$.
• Figure 2: The edge-corona product graph $P_{5}\diamond \overline{K_{3}}$.
• Figure 3: The local antimagic labeling of $F_{2,k}$ for $k\geq 3$.
• Figure 4: The local antimagic labeling of $F_{3,2}$.
• Figure 5: The local antimagic labeling of $F_{2,k}$.

Theorems & Definitions (23)

• Lemma 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• Theorem 3.3
• proof
• Corollary 3.1
• Theorem 3.4
• proof
• Theorem 3.5