Kemin Zhang

A note on list improper coloring planar graphs

Abstract A graph G is called (k, d)*-choosable if, for every list assignment L satisfying |L(v)| = k for all v ϵ V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this note, we prove that every planar graph without 4-cycles and l-cycles for some l ϵ {5, 6, 7} is (3, 1)*-choosable.

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.

A minimum broadcast graph on 26 vertices

Broadcasting is the process of information dissemination in a communication network in which a message, originated by one member, is transmitted to all members of the network. A broadcast graph is a graph which permits broadcasting from any originator in minimum time. The broadcast function B(n) is the minimum number of edges in any broadcast graph on n vertices. In this paper, we construct a broadcast graph on 26 vertices with 42 edges to prove B(26) = 42.

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 .

New Upper Bounds for Ramsey Numbers

The Ramsey numberR(G1,G2) is the smallest integerpsuch that for any graphGonpvertices eitherGcontainsG1orGcontainsG2, whereGdenotes the complement ofG. LetR(m,n)=R(Km,Kn). Some new upper bound formulas are obtained forR(G1,G2andR(m,n), and we derive some new upper bounds for Ramsey numbers here.

Pancyclic out-arcs of a vertex in tournaments

Abstract Thomassen (J. Combin. Theory Ser. B 28, 1980 , 142–163) proved that every strong tournament contains a vertex x such that each arc going out from x is contained in a Hamiltonian cycle. In this paper, we extend the result of Thomassen and prove that a strong tournament contains a vertex x such that every arc going out from x is pancyclic, and our proof yields a polynomial algorithm to find such a vertex. Furthermore, as another consequence of our main theorem, we get a result of Alspach (Canad. Math. Bull. 10, 1967 , 283–286) that states that every arc of a regular tournament is pancyclic.

The local exponent sets of primitive digraphs

Abstract Let D=(V,E) be a primitive digraph. The local exponent of D at a vertex u∈V , denoted by exp D (u) , is defined to be the least integer k such that there is a directed walk of length k from u to v for each v∈V . Let V={1,2,… , n} . The vertices of V can be ordered so that exp D (1)⩽ exp D (2)⩽⋯⩽ exp D (n)=γ(D) . We define the k th local exponent set E n (k):={ exp D (k)∣D∈PD n } , where PD n is the set of all primitive digraphs of order n . It is known that E n (n)={γ(D)∣D∈PD n } has been completely settled by K. Zhang [Linear Algebra Appl. 96 (1987) 102–108]. In 1998, E n (1) was characterized by J. Shen and S. Neufeld [Linear Algebra Appl. 268 (1998) 117–129]. In this paper, we describe E n (k) for all n,k with 2⩽k⩽n−1 . So the problem of local exponent sets of primitive digraphs is completely solved.

Complementary cycles containing a fixed arc in diregular bipartite tournaments

Abstract Let ( x , y ) be a specified arc in a k -regular bipartite tournament B . We prove that there exists a cycle C of length four through ( x , y ) in B such that B − C is hamiltonian.

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.

A bound for multicolor Ramsey numbers

Abstract The Ramsey number R(G 1 , G 2 , …, G n ) is the smallest integer p such that for any n -edge coloring (E 1 ,E 2 , …, E n ) of K p , K p [E i ] contains G i for some i , G i as a subgraph in K p [E i ] . Let R(m 1 ,m 2 , …, m n )≔R(K m 1 ,K m 2 , …, K m n ),R(m;n)≔R(m 1 ,m 2 ,…,m n ) if m i =m for i=1, 2,…,n . A formula is obtained for R(G 1 ,G 2 , …, G n ) .