An r-acyclic edge chromatic number of a graph G,denoted by α＇r（G）,is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C h...An r-acyclic edge chromatic number of a graph G,denoted by α＇r（G）,is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {｜C｜,r} colors.We prove that α＇r（G） ≤（4r ＋ 1）△（G）,when the girth of the graph G equals to max{50,△A（G）} and 4 ≤ r ≤ 7.If we relax the restriction of the girth to max {220,A（G）},the upper bound of a＇r（G） is not larger than（2r ＋ 5）△（G） with 4 ≤r≤ 10.展开更多
The Merrifield-Simmons index and Hosoya index are defined as the number of the graph G（V, E） as the number of subsets of V（G） in which no tow vertices are adjacent and the number of subsets of E（G） in which no t...The Merrifield-Simmons index and Hosoya index are defined as the number of the graph G（V, E） as the number of subsets of V（G） in which no tow vertices are adjacent and the number of subsets of E（G） in which no two edges are incident, respectively. In this paper, we characterize the Unicyclic graphs with Merrifield-Simmons indices and Hosoya indices, respectively. And double-cyclic graphs with Hosoya indices among the doublecyclic graphs with n vertices.展开更多
基金Supported by the National Natural Science Foundation of China(No.11371355)
文摘An r-acyclic edge chromatic number of a graph G,denoted by α＇r（G）,is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {｜C｜,r} colors.We prove that α＇r（G） ≤（4r ＋ 1）△（G）,when the girth of the graph G equals to max{50,△A（G）} and 4 ≤ r ≤ 7.If we relax the restriction of the girth to max {220,A（G）},the upper bound of a＇r（G） is not larger than（2r ＋ 5）△（G） with 4 ≤r≤ 10.
基金This project is supported by National Natural Science Foundation of China（10671081） and the Science Foundation of Hubei Province（2006AA412C27）
文摘The Merrifield-Simmons index and Hosoya index are defined as the number of the graph G（V, E） as the number of subsets of V（G） in which no tow vertices are adjacent and the number of subsets of E（G） in which no two edges are incident, respectively. In this paper, we characterize the Unicyclic graphs with Merrifield-Simmons indices and Hosoya indices, respectively. And double-cyclic graphs with Hosoya indices among the doublecyclic graphs with n vertices.
