In this paper, we present a new concept of the adjacent-vertex-distinguishing total coloring of graphs (briefly, AVDTC of graphs) and, meanwhile, have obtained the adjacent-vertex-distinguishing total chromatic number...In this paper, we present a new concept of the adjacent-vertex-distinguishing total coloring of graphs (briefly, AVDTC of graphs) and, meanwhile, have obtained the adjacent-vertex-distinguishing total chromatic number of some graphs such as cycle, complete graph, complete bipartite graph, fan, wheel and tree.展开更多
For any vertex u∈V（G）, let T<sub>N</sub>（U）={u}∪{uv|uv∈E（G）, v∈v（G）}∪{v∈v（G）|uv∈E（G）}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C&...For any vertex u∈V（G）, let T<sub>N</sub>（U）={u}∪{uv|uv∈E（G）, v∈v（G）}∪{v∈v（G）|uv∈E（G）}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C<sub>f</sub>（u）={f（x）|x∈TN（U）}. For any two adjacent vertices x and y of V（G）such that C<sub>f</sub>（x）≠C<sub>f</sub>（y）, we refer to f as a k-avsdt-coloring of G（'avsdt'is the abbreviation of'adjacent-vertex-strongly- distinguishing total'）. The avsdt-coloring number of G, denoted by X<sub>ast</sub>（G）, is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ（G）+1≤X<sub>ast</sub>（G）≤Δ（G）+2 for any tree or unique cycle graph G.展开更多
A new concept of the D(β)-vertex-distinguishing total coloring of graphs, i.e., the proper total coloring such that any two vertices whose distance is not larger than β have different color sets, where the color set...A new concept of the D(β)-vertex-distinguishing total coloring of graphs, i.e., the proper total coloring such that any two vertices whose distance is not larger than β have different color sets, where the color set of a vertex is the set composed of all colors of the vertex and the edges incident to it, is proposed in this paper. The D(2)-vertex-distinguishing total colorings of some special graphs are discussed, meanwhile, a conjecture and an open problem are presented.展开更多
文摘In this paper, we present a new concept of the adjacent-vertex-distinguishing total coloring of graphs (briefly, AVDTC of graphs) and, meanwhile, have obtained the adjacent-vertex-distinguishing total chromatic number of some graphs such as cycle, complete graph, complete bipartite graph, fan, wheel and tree.
基金the National Natural Science Foundation of China (Grant Nos. 10771091, 10661007)
文摘For any vertex u∈V（G）, let T<sub>N</sub>（U）={u}∪{uv|uv∈E（G）, v∈v（G）}∪{v∈v（G）|uv∈E（G）}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C<sub>f</sub>（u）={f（x）|x∈TN（U）}. For any two adjacent vertices x and y of V（G）such that C<sub>f</sub>（x）≠C<sub>f</sub>（y）, we refer to f as a k-avsdt-coloring of G（'avsdt'is the abbreviation of'adjacent-vertex-strongly- distinguishing total'）. The avsdt-coloring number of G, denoted by X<sub>ast</sub>（G）, is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ（G）+1≤X<sub>ast</sub>（G）≤Δ（G）+2 for any tree or unique cycle graph G.
文摘A new concept of the D(β)-vertex-distinguishing total coloring of graphs, i.e., the proper total coloring such that any two vertices whose distance is not larger than β have different color sets, where the color set of a vertex is the set composed of all colors of the vertex and the edges incident to it, is proposed in this paper. The D(2)-vertex-distinguishing total colorings of some special graphs are discussed, meanwhile, a conjecture and an open problem are presented.
