Let G : Gn,p be a binomial random graph with n vertices and edge probability p = p(n), and f be a nonnegative integer-valued function defined on V(G) such that 0 〈 a ≤ f(x) ≤ b 〈 np- 2√nplogn for every ...Let G : Gn,p be a binomial random graph with n vertices and edge probability p = p(n), and f be a nonnegative integer-valued function defined on V(G) such that 0 〈 a ≤ f(x) ≤ b 〈 np- 2√nplogn for every E V(G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0, 1] so that for each vertex x, we have d^hG(x) = f(x), where dh(x) = ∑ h(e) is the fractional degree xEe ofx inG. Set Eh = {e : e e E(G) and h(e) ≠ 0}. IfGh isaspanningsubgraphofGsuchthat E(Gh) = Eh, then Gh is called an fractional f-factor of G. In this paper, we prove that for any binomial random graph Gn,p 2 with p 〉 n^-2/3, almost surely Gn,p contains an fractional f-factor.展开更多
A vertex coloring of a graph G is called r-acyclic if it is a proper vertex coloring such that every cycle D receives at least min{|D|, r} colors. The r-acyclic chromatic number of G is the least number of colors in...A vertex coloring of a graph G is called r-acyclic if it is a proper vertex coloring such that every cycle D receives at least min{|D|, r} colors. The r-acyclic chromatic number of G is the least number of colors in an r-acyclic coloring of G. We prove that for any number r ≥ 4, the r-acyclic chromatic number of any graph G with maximum degree △ ≥ 7 and with girth at least(r-1)△ is at most(4r-3)△.展开更多
基金Supported by NSFSD(No.ZR2013AM001)NSFC(No.11001055),NSFC11371355
文摘Let G : Gn,p be a binomial random graph with n vertices and edge probability p = p(n), and f be a nonnegative integer-valued function defined on V(G) such that 0 〈 a ≤ f(x) ≤ b 〈 np- 2√nplogn for every E V(G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0, 1] so that for each vertex x, we have d^hG(x) = f(x), where dh(x) = ∑ h(e) is the fractional degree xEe ofx inG. Set Eh = {e : e e E(G) and h(e) ≠ 0}. IfGh isaspanningsubgraphofGsuchthat E(Gh) = Eh, then Gh is called an fractional f-factor of G. In this paper, we prove that for any binomial random graph Gn,p 2 with p 〉 n^-2/3, almost surely Gn,p contains an fractional f-factor.
基金supported by NSFC(Grant number 11571258)SDNSF(Grant number ZR2016AM01)
文摘A vertex coloring of a graph G is called r-acyclic if it is a proper vertex coloring such that every cycle D receives at least min{|D|, r} colors. The r-acyclic chromatic number of G is the least number of colors in an r-acyclic coloring of G. We prove that for any number r ≥ 4, the r-acyclic chromatic number of any graph G with maximum degree △ ≥ 7 and with girth at least(r-1)△ is at most(4r-3)△.