Yunqing Zhang

The Ramsey numbers of stars versus wheels

For two given graphs G 1 and G 2 , the Ramsey number R(G 1 , G 2 ) is the smallest positive integer n such that for any graph G of order n, either G contains G 1 or the complement of G contains G 2 . Let S n denote a star of order n and W m a wheel of order m + 1. This paper shows that R(S n , W 6 )=2n + 1 for n ≥ 3 and R(S n , W m )=3n - 2 for m odd and n ≥ m - 1 ≥ 2.

The Ramsey numbers of paths versus wheels

Abstract For two given graphs G 1 and G 2 , the Ramsey number R ( G 1 , G 2 ) is the smallest integer n such that for any graph G of order n, either G contains G 1 or the complement of G contains G 2 . Let P n denote a path of order n and W m a wheel of order m + 1 . In this paper, we show that R ( P n , W m ) = 2 n - 1 for m even and n ⩾ m - 1 ⩾ 3 and R ( P n , W m ) = 3 n - 2 for m odd and n ⩾ m - 1 ⩾ 2 .

The Ramsey numbers of trees versus W 6 or W 7

Let Tn denote a tree of order n and Wm a wheel of order m + 1. In this paper, we show the Ramsey numbers R(Tn, W6) = 2n - 1 + µ for n ≥ 5, where µ = 2 if Tn = Sn, µ = 1 if Tn = Sn(1, 1) or Tn = Sn(1, 2) and n ≡ 0 (mod 3), and µ = 0 otherwise; R(Tn, W7) = 3n - 2 for n ≥ 6.

The Ramsey numbers R(Cm,K7) and R(C7,K8)

For two given graphs G1G1 and G2G2, the Ramsey number R(G1,G2)R(G1,G2) is the smallest integer nn such that for any graph GG of order nn, either GG contains G1G1 or the complement of GG contains G2G2. Let CmCm denote a cycle of length mm and KnKn a complete graph of order nn. In this paper we show that R(Cm,K7)=6m−5R(Cm,K7)=6m−5 for m≥7m≥7 and R(C7,K8)=43R(C7,K8)=43, with the former result confirming a conjecture due to Erdos, Faudree, Rousseau and Schelp that R(Cm,Kn)=(m−1)(n−1)+1R(Cm,Kn)=(m−1)(n−1)+1 for m≥n≥3m≥n≥3 and (m,n)≠(3,3)(m,n)≠(3,3) in the case where n=7n=7.

THE RAMSEY NUMBERS FOR STARS OF ODD ORDER VERSUS A WHEEL OF ORDER NINE

For two given graphs G1 and G2, the Ramsey number R(G1, G2) is the smallest positive integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2. Let Sn denote a star of order n and Wm a wheel of order m + 1. In this paper we show that R(Sn, W8) = 2n + 1 for n ≥ 5 and n ≡ 1 (mod 2).

The Ramsey numbers R(Tn, W6) for Δ(Tn) ≥ n−3☆

Abstract Let Tn denote a tree of order n and Wm a wheel of order m + 1. In a previous paper, we evaluated the Ramsey number R(Tn, Wm) in the cases where Tn is the star of order n and m = 6 or m is odd and n ≥ m − 1 ≥ 2. In this paper, we determine R(Tn, W6) in the case where the maximum degree of Tn is at least n − 3. Our results show that a recent conjecture of Baskoro et al. is false.

An Ore-type Condition for Cyclability

A graph G is said to be cyclable if for each orientation D of G, there exists a set S(D) ?V(G) such that reversing all the arcs with one end in S results in a Hamiltonian digraph. Let G be a simple graph of even ordern? 8. In this paper, we show that if the degree sum of any two nonadjacent vertices is not less thann+ 1, then G is cyclable and the lower bound is sharp.

Carbon-constrained perishable inventory management with freshness-dependent demand

We consider perishable inventory control with freshness-dependent demand under carbon emissions constraints. We propose two deteriorating inventory models with carbon emissions tax and the capand-trade mechanism, in which the demand is freshness dependent, carbon emissions come from inventory holding, shipping, and item deteriorating, and the objective is to maximize the profit per unit time. We characterize the existence and uniqueness of the solutions for the models. We analyse the impacts of carbon emissions tax, carbon emissions quota, and carbon price on inventory decisions, carbon emissions, and profit. We conduct simulation to generate managerial insights from our analytical results. (Received, processed, and accepted by the Chinese Representative Office.)

The Ramsey numbers R(Cm,K7)R(Cm,K7) and R(C7,K8)R(C7,K8)

For two given graphs G1G1 and G2G2, the Ramsey number R(G1,G2)R(G1,G2) is the smallest integer nn such that for any graph GG of order nn, either GG contains G1G1 or the complement of GG contains G2G2. Let CmCm denote a cycle of length mm and KnKn a complete graph of order nn. In this paper we show that R(Cm,K7)=6m−5R(Cm,K7)=6m−5 for m≥7m≥7 and R(C7,K8)=43R(C7,K8)=43, with the former result confirming a conjecture due to Erdos, Faudree, Rousseau and Schelp that R(Cm,Kn)=(m−1)(n−1)+1R(Cm,Kn)=(m−1)(n−1)+1 for m≥n≥3m≥n≥3 and (m,n)≠(3,3)(m,n)≠(3,3) in the case where n=7n=7.

The Ramsey numbers R ( C m , K 7 ) and R ( C 7 , K 8 )

For two given graphs G1G1 and G2G2, the Ramsey number R(G1,G2)R(G1,G2) is the smallest integer nn such that for any graph GG of order nn, either GG contains G1G1 or the complement of GG contains G2G2. Let CmCm denote a cycle of length mm and KnKn a complete graph of order nn. In this paper we show that R(Cm,K7)=6m−5R(Cm,K7)=6m−5 for m≥7m≥7 and R(C7,K8)=43R(C7,K8)=43, with the former result confirming a conjecture due to Erdos, Faudree, Rousseau and Schelp that R(Cm,Kn)=(m−1)(n−1)+1R(Cm,Kn)=(m−1)(n−1)+1 for m≥n≥3m≥n≥3 and (m,n)≠(3,3)(m,n)≠(3,3) in the case where n=7n=7.