mC₃∨nC₃和mC₄∨nC₄点可区别Ⅰ-全染色及Ⅵ-全染色0107-04mC₃∨nC₃和mC₄∨nC₄点可区别 Ⅰ-全染色及Ⅵ-全染色

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). Let χ_vt ^Ⅰ(G)=min{k|G has a k-VDIT coloring}, then χ_vt ^Ⅰ(G) is called the VDIT chromatic number of graph G. Let χ_vt ^Ⅵ(G)=min{k|G has a k-VDVIT coloring}, then χ_vt ^Ⅵ(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.