Chen Xiang-en

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u ∊ V(G), let TN(u) = {u} ∪ {uυ|uυ ∊ E(G), υ ∊ υ(G)} ∪ {υ ∊ υ(G)|uυ ∊ 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 Cf(u) = {f(x) | x ∊ TN(u)}. For any two adjacent vertices x and y of V(G) such that Cf(x) ≠ Cf(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 χast(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 prove Δ(G) + 1 ⩽ χast(G) ⩽ Δ(G) + 2 for any tree or unique cycle graph G.

Labelling Sun-Like Graphs from Scale-Free Small-World Network Models

General sun-graphs are applied to an actual ring network. Each node in the network represents a server, which is equivalent to that the graph is connected by nodes represented as servers. One can use labellings to distinguish nodes and edges between nodes in order to find some fast algorithms to imitate some effective transmissions and communications in information networks. We propose method for constructing scale-free small-world network models, also, building sun-like network models, motivated from some ring real-networks, and show that sun-like network models have can be strictly distinguished by felicitous labellings.