Yaojun Chen

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 for cycles versus wheels of odd order

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 C n denote a cycle of order n and W m a wheel of order m + 1 . It is conjectured by Surahmat, E.T. Baskoro and I. Tomescu that R ( C n , W m ) = 2 n − 1 for even m ≥ 4 , n ≥ m and ( n , m ) ≠ ( 4 , 4 ) . In this paper, we confirm the conjecture for n ≥ 3 m / 2 + 1 .

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.

Polarity graphs and Ramsey numbers for C 4 versus stars

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 of order N , either G contains a copy of G 1 or its complement contains a copy of G 2 . Let C m be a cycle of length m and K 1 , n a star of order n + 1 . Parsons (1975) shows that R ( C 4 , K 1 , n ) ź n + ź n - 1 ź + 2 and if n is the square of a prime power, then the equality holds. In this paper, by discussing the properties of polarity graphs whose vertices are points in the projective planes over Galois fields, we prove that R ( C 4 , K 1 , q 2 - t ) = q 2 + q - ( t - 1 ) if q is an odd prime power, 1 ź t ź 2 ź q 4 ź and t ź 2 ź q 4 ź - 1 , which extends a result on R ( C 4 , K 1 , q 2 - t ) obtained by Parsons (1976).

On star-critical and upper size Ramsey numbers

In this paper, we study the upper size Ramsey number u ( G 1 , G 2 ) , defined by Erd?s and Faudree, as well as the star-critical Ramsey number r ? ( G 1 , G 2 ) , defined by Hook and Isaak. We define Ramsey-full graphs and size Ramsey good graphs, and perform a detailed study on these graphs. We generalize earlier results by determining u ( n K k , m K l ) and r ? ( n K k , m K l ) for k , l ? 3 and large m , n ; u ( C n , C m ) for m odd, with n m ? 3 ; and r ? ( C n , C m ) for m odd, with n ? m ? 3 and ( m , n ) ? ( 3 , 3 ) .

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.

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.