The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs

Xue Yang,Haizheng Yu,Hong Bian,Dandan Liu

doi:10.3390/axioms11030097

Abstract

Let G=(V,E) be a connected graph with |V|=n and |E|=m. A bijection f:V(G)∪E(G)→{1,2,⋯,n+m} is called local antimagic total labeling if, for any two adjacent vertices u and v, ωt(u)≠ωt(v), where ωt(u)=f(u)+∑e∈E(u)f(e), and E(u) is the set of edges incident to u. Thus, any local antimagic total labeling induces a proper coloring of G, where the vertex x in G is assigned the color ωt(x). The local antimagic total chromatic number, denoted by χlat(G), is 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 numbers of some wheel-related graphs, such as the fan graph Fn, the bowknot graph Bn,n, the Dutch windmill graph D4n, the analogous Dutch graph AD4n and the flower graph Fn.

Highlights

Let G = (V, E) be a connected simple graph with n vertices and m edges
We present the local antimagic total chromatic numbers of some wheel-related graphs, such as the fan graph Fn, the bowknot graph Bn,n, the Dutch windmill graph D4n, the analogous Dutch graph AD4n and the flower graph Fn
From the results proved by Haslegrave [9], Baca et al obtained that the local antimagic chromatic numbers of disjoint union of arbitrary graphs are finite if and only if none of these graphs contain an isolated edge as a subgraph

Summary

Introduction

The local antimagic total labeling on a graph G is defined to be an assignment f : V(G) ∪ E(G) → {1, 2, · · · , |V(G)| + |E(G)|} so that the weights of any two adjacent vertices u and v are distinct, that is, ωt(u) = ωt(v), where ωt(u) = f (u) + ∑e∈E(u) f (e). The following local antimagic total labeling of the fan graph in Theorem 1 is different from that of these authors.

Results

Conclusion