2-Distance sum distinguishing total coloring of Halin graphs
Let[k]={1,2,…,k} be a color set.Let f:V(G)U E(G)→[k]be a k-total coloring of G.Set S(u)=f(u)+∑v∈NG(u)f(uv),where NG(u)is the neighbor set of vertex u.If S(u)≠ S(v)for any two vertices u,v with their distance is not more than 2,then f is called the 2-distance sum distinguishing k-total coloring of G.The smallest valuek that G admits a 2-distance sum distinguishing k-total coloring of G is called the 2-distance sum distinguishing total chromatic number of G,and denoted by x"2-∑(G).By using Combinatorial Nullstellensatz,it is proved that x"2-∑(G)≤max{△(G)+2,9} for Halin graph G with maximum degree △(G)≥4.
2-distance sum distinguishing total coloringHalin graphCombinatorial Nullstellensatz
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