Local antimagic chromatic number of blossomed star
[Objective]As a connected simple graph,graphG=(V(G),E(G))is followed by|V(G)|=n,|E(G)|=m.In this paper,the local antimagic coloring of graphs is studied.[Methods]A bijectionf:E(G)→ {1,2,…,m} is called a local antimagic labeling of G such that ω(u)≠ω(v)for any two adjacent vertices u and v in G,where ω(u)=∑e∈E(u)f(e),and E(u)is the set of edges incident to u.Clearly,any local antimagic labeling induces a proper vertex coloring of G,where the vertex x is assigned the colorω(x).The local antimagic chromatic number,denoted byxla(G),is the minimum number of colors taken over all colorings induced by the local antimagic labelings of G.A tree T is called a blossomed star graph if it can be obtained by adding some pendant edges to the pendant vertices of a star tree.Let Tn,m denote the blossomed star obtained by adding m pendant edges to every pendant vertex of the star tree S1,n.The local antimagic coloring is applied to the blossomed star graph,and the local antimagic labeling is determined step by step according to the structure of the blossomed star.Finally,the color number close to the number of pendant edges of the blossomed star is used as far as possible to obtain the corresponding local antimagic chromatic number.[Results]For the blossomed star Tn,m,we find that(1)Xla(Tn,m)=mn+2 when n＞2m+1;or n=2,evenm(m≥4);(2)Xla(Tn,m)=mn+1 when m is odd and n≤2m+1;or m=2,n=2,4;or even m,odd n and n≤2m+1;or even m,n(m,n≥4)and n=m,m+2;(3)mn+1 ≤ Xla(Tn,m)≤mn+2 whenm,n(m≥4,n≥4)are even and n≠m,m+2.[Conclusion]The local antimagic coloring is performed on the blossomed star graph and the local antimagic chromatic number is obtained.
local antimagic labelinglocal antimagic chromatic numberblossomed star
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