mC3∨nC3和mC4∨nC4点可区别Ⅰ-全染色及Ⅵ-全染色
Vertex-distinguishingⅠ-total coloring and Ⅵ-total coloring ofmC3∨nC3 andmC4∨nC4

DOI：10.7511/dllgxb202001015

 作者 单位 陈祥恩,张生桂

设f为简单图G的一个一般全染色(即若干种颜色对图G的全部顶点及边的一个分配)，如果任意两个相邻点染以不同颜色且任意两条相邻边染以不同的颜色，则称为图G的Ⅰ-全染色；如果任意两条相邻边染以不同的颜色，则称为图G的Ⅵ-全染色．用C(x)表示在f下点x的颜色以及与x关联的边的色所构成的集合(非多重集)．对图G的一个Ⅰ-全染色(分别地，Ⅵ-全染色)f，一旦u，v∈V(G)，u≠v，就有C(u)≠C(v)，则f称为图G的点可区别Ⅰ-全染色(或点可区别Ⅵ-全染色)，简称为VDIT染色(分别地，VDVIT染色)．令χⅠvt(G)=min{k|G存在k-VDIT染色}，称χⅠvt(G)为图G的点可区别Ⅰ-全色数．令χⅥ vt(G)=min{k|G存在k-VDVIT染色}，称χⅥvt(G)为图G的点可区别Ⅵ-全色数．利用构造具体染色的方法，讨论了联图mC3∨nC3和mC4∨nC4的点可区别Ⅰ-全染色和点可区别Ⅵ-全染色，并给出了联图mC3∨nC3和mC4∨nC4的点可区别Ⅰ-全色数和点可区别Ⅵ-全色数．

Let Gbe a simple graph. Suppose fis a general total coloring of graph G(i.e., an assignment of several colors to all vertices and edges of G), if any two adjacent vertices and any two adjacent edges of graph Gare assigned different colors, then fis called an Ⅰ-total coloring of a graph G; if any two adjacent edges of Gare assigned different colors, thenf is called a Ⅵ-total coloring of a graph G. Let C(x) denote the set of colors of vertex x and of the edges incident with x under f, the set is non multiple set. For an Ⅰ-total coloring (resp., Ⅵ-total coloring) f of a graph G, if C(u)≠C(v) for any two distinct vertices u and v of V(G), then fis called a vertex-distinguishing Ⅰ-total coloring (resp., vertex-distinguishing Ⅵ-total coloring) of graph G, short for VDIT coloring (resp., VDVIT coloring). LetχⅠvt(G)=min{k|G has ak-VDIT coloring}, then χⅠvt(G) is called the VDIT chromatic number of graph G. Let χⅥ vt(G)=min{k|G has a k-VDVIT coloring}, then χⅥvt(G) is called the VDVIT chromatic number of graph G. The VDIT coloring (resp., VDVIT coloring) of mC3∨nC3 andmC4∨nC4are determined and the VDIT chromatic number (resp., VDVIT chromatic number) of them are determined by constructing concrete coloring.