The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs - Xue Yang; Hong Bian; Haizheng Yu; Dandan Liu (Axioms, 2022)
Let <semantics>G=(V,E)</semantics> be a connected graph with <semantics>|V|=n</semantics> and <semantics>|E|=m</semantics>. A bijection <semantics>f:V(G)∪E(G)→{1,2,⋯,n+m}</semantics> is called local antimagic total labeling if, for any two adjacent vertices u and v, <semantics>ωt(u)≠ωt(v)</semantics>, where <semantics>ωt(u)=f(u)+∑e∈E(u)f(e)</semantics>, and <semantics>E(u)</semantics> 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 <semantics>ωt(x)</semantics>. The local antimagic total chromatic number, denoted by <semantics>χlat(G)</semantics>, 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 <semantics>Fn</semantics>, the bowknot graph <semantics>Bn,n</semantics>, the Dutch windmill graph <semantics><msubsup>D4n</msubsup></semantics>, the analogous Dutch graph <semantics>A<msubsup>D4n</msubsup></semantics> and the flower graph <semantics>Fn</semantics>.