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On local antimagic total labeling of complete graphs amalgamation

Lau Gee-Choon, Shiu Wai-Chee

Opuscula Mathematica

2023

Vol. 43, no. 3

429--453

Let G = (V,E) be a connected simple graph of order p and size q. A graph G is called local antimagic (total) if G admits a local antimagic (total) labeling. A bijection g : E → {1, 2, . . . , q} is called a local antimagic labeling of G if for any two adjacent vertices u and v, we have g+(u) ̸= g+(v), where g+(u) = ∑e∈E(u) g(e), and E(u) is the set of edges incident to u. Similarly, a bijection f : V (G)∪E(G) → {1, 2, . . . , p+q} is called a local antimagic total labeling of G if for any two adjacent vertices u and v, we have wf (u) ̸= wf (v), where wf (u) = f(u) + ∑e∈E(u) f(e). Thus, any local antimagic (total) labeling induces a proper vertex coloring of G if vertex v is assigned the color g+(v) (respectively, wf (u)). The local antimagic (total) chromatic number, denoted χla(G) (respectively χlat(G)), is the minimum number of induced colors taken over local antimagic (total) labeling of G. In this paper, we determined χlat(G) where G is the amalgamation of complete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of K1 and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.

Every graph is local antimagic total and its applications

Lau Gee-Choon, Schaffer Karl, Shiu Wai-Chee

Opuscula Mathematica

2023

Vol. 43, no. 6

841--864

Let G = (V,E) be a simple graph of order p and size q. A graph G is called local antimagic (total) if G admits a local antimagic (total) labeling. A bijection g : E → {1, 2, . . . , q} is called a local antimagic labeling of G if for any two adjacent vertices u and v, we have g+(u)≠g+(v), where g+(u) = ∑e∈E(u) g(e), and E(u) is the set of edges incident to u. Similarly, a bijection f : V (G)∪E(G) → {1, 2, . . . , p+q} is called a local antimagic total labeling of G if for any two adjacent vertices u and v, we have wf (u)≠wf (v), where wf (u) = f(u) + ∑e∈E(u) f(e). Thus, any local antimagic (total) labeling induces a proper vertex coloring of G if vertex v is assigned the color g+(v) (respectively, wf (u)). The local antimagic (total) chromatic number, denoted χla(G) (respectively χlat(G)), is the minimum number of induced colors taken over local antimagic (total) labeling of G. We provide a short proof that every graph G is local antimagic total. The proof provides sharp upper bound to χlat(G). We then determined the exact χlat(G), where G is a complete bipartite graph, a path, or the Cartesian product of two cycles. Consequently, the χla(G ∨ K1) is also obtained. Moreover, we determined the χla(G ∨ K1) and hence the χlat(G) for a class of 2-regular graphs G (possibly with a path). The work of this paper also provides many open problems on χlat(G). We also conjecture that each graph G of order at least 3 has χlat(G) ≤ χla(G).