In this paper we essentially determine all covers {a s (mod n s)} k s=1 of Z with k<10 , actually our algorithm is valid for any positive integer k . As an application we provide a somewhat ge...In this paper we essentially determine all covers {a s (mod n s)} k s=1 of Z with k<10 , actually our algorithm is valid for any positive integer k . As an application we provide a somewhat general theorem on (infinite) arithmetic progressions (e.g. 1330319+346729110 Z) consisting of odd integers no term of which can be expressed as the sum of a power of two and an odd prime, on the other hand we obtain an interesting result on integers of the form 2 n+cp where c is a constant and p is a prime.展开更多
A Hamiltonian k-factor is a k-factor containing aHamiltonian cycle.An n/2-critical graph G is a simple graph of order n which satisfies δ(G)≥n/2 and δ(G-e)<n/2 for any edge e∈E(G).Let k≥2 be an integer and G b...A Hamiltonian k-factor is a k-factor containing aHamiltonian cycle.An n/2-critical graph G is a simple graph of order n which satisfies δ(G)≥n/2 and δ(G-e)<n/2 for any edge e∈E(G).Let k≥2 be an integer and G be an n/2-critical graph of even order n≥8k-14.It is shown in this paper that for any given Hamiltonian cycle C except that G-C consists of two components of odd orders when k is odd,G has a k-factor containing C.展开更多
文摘In this paper we essentially determine all covers {a s (mod n s)} k s=1 of Z with k<10 , actually our algorithm is valid for any positive integer k . As an application we provide a somewhat general theorem on (infinite) arithmetic progressions (e.g. 1330319+346729110 Z) consisting of odd integers no term of which can be expressed as the sum of a power of two and an odd prime, on the other hand we obtain an interesting result on integers of the form 2 n+cp where c is a constant and p is a prime.
基金This research is supported partially by the National Natural Science Foundation of China.
文摘A Hamiltonian k-factor is a k-factor containing aHamiltonian cycle.An n/2-critical graph G is a simple graph of order n which satisfies δ(G)≥n/2 and δ(G-e)<n/2 for any edge e∈E(G).Let k≥2 be an integer and G be an n/2-critical graph of even order n≥8k-14.It is shown in this paper that for any given Hamiltonian cycle C except that G-C consists of two components of odd orders when k is odd,G has a k-factor containing C.
