The local antimagic (total) chromatic numbers of firecracker graphs and edge-corona product graphs
College of Mathematics and System Sciences, Xinjiang University（新疆大学数学与系统科学学院）；School of Mathematical Sciences Xinjiang Normal University（新疆师范大学数学科学学院）
Xue Yang, Hong Bian, Xueliang Li, Zhixia Yang, Haizheng Yu
设G=(V(G),E(G))是一个具有n个顶点和m条边的连通简单图。将边集映射到[m]的函数称为G的局部反魔法标记，如果对于任意两个相邻顶点u和v，与u和v关联的边的权重和不同。同样，将边集和顶点集的并集映射到[n+m]的函数称为G的局部反魔法总标记，如果对于任意两个相邻顶点u和v，u的权重和其相邻边的权重之和与v的不同。显然，任何局部反魔法（总）标记在将每个顶点v分配颜色w(v)(w_t(v))时都会诱导一个正常的顶点着色。G的局部反魔法（总）色数，记为X_la(G)(X_lat(G))，定义为所有由局部反魔法（总）标记诱导的颜色数的最小值。在本文中，我们给出了火柴人图F_n,k的局部反魔法（总）色数，F_n,k由n个k星通过连接每个星的一个叶来拼接得到。然后我们给出了两个图G和H的边冠状积图的局部反魔法色数，其中该图通过取一个G的副本和|E(G)|个互不相交的H副本，将每个H副本一一对应地分配给G的每条边，对于每条边uv of G，将u和v连接到与uv关联的H副本的每个顶点。对于本文研究的图，G是一个星S_k或双星S_k1,k2，而H是一个具有r个顶点的空图或一个完全图K_2。
英文摘要：
Let G=(V(G),E(G)) be a connected simple graph with n vertices and m edges. A bijection f from the edge set of G to [m] is called a local antimagic labeling of G, if for any two adjacent vertices u and v in G, the sums of the weights of the edges associated with u and v ,respectively, are different. Similarly, A bijection g from the union of edge set and vertex set of G to [n+m] is called a local antimagic total labeling of G, if for any two adjacent vertices u and v in G, The sum of the weight of u and the weights of its incident edges differs from that of v. Obviously, any local antimagic (total) labeling induces a proper vertex-coloring of G when every vertex v is assigned the color w(v)(w_t(v)). The local antimagic (total) chromatic number of G, denoted by X_la(G)(X_lat(G)) , is defined as 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 number of firecracker graph F_n,k, obtained by the concatenation of n k-stars by linking one leaf from each. Then we give the local antimagic chromatic number of the edge-corona product of two graphs G and H, where the graph is constructed by taking one copy of G and |E(G)| disjoint copies of H one-to-one assigned to each edge of G, and for every edge uv of G, joining u and v to every vertex of the copy of H associated to uv. For the graph studied here, G is a star S_k or a double star S_k1,k2, and H is an empty graph with r vertices or a complete graph K_2.
On local antimagic chromatic number of a corona product graph
本文给出了[On the local antimagic vertex coloring of corona products related to friendship and fan graph，{ \it Indon. J. Combin.}，5（2）（2021）110- 121]。从而得到了友谊图和零图的晕积的精确局部反幻色数的一个精确下界。
