For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime...For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime divisors of n. Some kind of equations involving Euler＇s function is studied in the paper.展开更多
Let G be a graph, and g and f be integer valued functions defined on V(G) which satisfy g(x)≤f(x) and g(x)≡f(x)(mod 2) for all x∈V(G). Then a spanning subgraph F of G is called a {g,g+2,…,f} -factor if deg_F(x)∈{...Let G be a graph, and g and f be integer valued functions defined on V(G) which satisfy g(x)≤f(x) and g(x)≡f(x)(mod 2) for all x∈V(G). Then a spanning subgraph F of G is called a {g,g+2,…,f} -factor if deg_F(x)∈{g(x),g(x)+2,…,f(x)} for all x∈V(G), when g(x)=1 for all x∈V(G), such a factor is called (1,f) -odd-factor. We give necessary and sufficient conditions for a graph G to have a {g,g+2,…,f} -factor and a (1,f) -odd-factor which contains an arbitrarily given edge of G, from that we derive some interesting results.展开更多
基金Foundation item： Supported by the National Natural Science Foundation of China（10671056）
文摘For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime divisors of n. Some kind of equations involving Euler＇s function is studied in the paper.
文摘Let G be a graph, and g and f be integer valued functions defined on V(G) which satisfy g(x)≤f(x) and g(x)≡f(x)(mod 2) for all x∈V(G). Then a spanning subgraph F of G is called a {g,g+2,…,f} -factor if deg_F(x)∈{g(x),g(x)+2,…,f(x)} for all x∈V(G), when g(x)=1 for all x∈V(G), such a factor is called (1,f) -odd-factor. We give necessary and sufficient conditions for a graph G to have a {g,g+2,…,f} -factor and a (1,f) -odd-factor which contains an arbitrarily given edge of G, from that we derive some interesting results.
