On Some Three-Color Ramsey Numbers

 In this paper we study three-color Ramsey numbers. Let K _i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G ₁, G ₂ and G ₃, if r(G ₁, G ₂)≥s(G ₃), then r(G ₁, G ₂, G ₃)≥(r(G ₁, G ₂)−1)(χ(G ₃)−1)+s(G ₃), where s(G ₃) is the chromatic surplus of G ₃; (ii) (k+m−2)(n−1)+1≤r(K _1,k , K _1,m , K _n )≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m≥k≥2, there is a constant c such that r(K _k,m , K _k,m , K _n )≤c(n/logn), and r(C _2m , C _2m , K _n )≤c(n/logn) ^m/(m−1) for sufficiently large n. Received: July 25, 2000 Final version received: July 30, 2002 RID="*" ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of education ministry of China, and the foundations of Jiangsu Province Acknowledgments. The authors are grateful to the referee for his valuable comments. AMS 2000 MSC: 05C55     

The girth of a 4-homogeneous bipartite graph

In this paper, it is proved that the girth of a 4-homogeneous bipartite graph with valency greater than 2 is at most 12.     

Graph Automorphisms and Cells of Lattices

In this paper we apply the notion of cell of a lattice for dealing with graph automorphisms of lattices (in connection with a problem proposed by G. Birkhoff).     

Bound on <Emphasis Type="Italic">m</Emphasis>-restricted Edge Connectivity

An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restrict edge connectivity λm is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let θ(X) denote the number of edges with one end in X and the other not in X and ξm=min{θ(X) ;X is a connected vertex-induced subgraph of order m}.It is proved in this paper that if G has girth at least m/2 2,then λm≤ξm.The upper bound of λm is sharp.     

Disjoint Paths in Graphs III,Characterization

Let G be a graph, $ \{a, b, c\}\subseteq V(G) $, and $ \{a', b', c'\}\subseteq V(G) $ such that $ \{a, b, c\}\neq \{a', b', c'\} $. We say that $ (G, \{a, c\}, \{a', c'\}, (b, b')) $ is an obstruction if, for any three vertex disjoint paths from {a, b, c} to {a', b', c'} in G, one path is from b to b'. In this paper we characterize obstructions. As a consequence, we show that no obstruction can be 8-connected, unless b = b' or {a, c} = {a', c'}.AMS Subject Classification: 05C38.     

Sparse pseudo‐random graphs are Hamiltonian

In this article we study Hamilton cycles in sparse pseudo‐random graphs. We prove that if the second largest absolute value λ of an eigenvalue of a d‐regular graph G on n vertices satisfies and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for every c >0 and large enough n a Cayley graph X (G,S), formed by choosing a set S of c log⁵ n random generators in a group G of order n, is almost surely Hamiltonian. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 17–33, 2003     

Maximal <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>‐regularity for elliptic operators with VMO‐coefficients

In this paper we show that the parabolic problem associated with elliptic operators A having coefficients in has the property of maximal -regularity. RID="h1" ID="h1"In Memoriam Philippe Bénilan     

Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes

We define an analogue of Schnyder's tree decompositions for 3-connected planar graphs. Based on this structure we obtain: Let G be a 3-connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f–1)×(f–1) grid. Let G be a 3-connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3.The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.     

Analysis of planar waveguides with index discontinuity or index slope discontinuity

In this paper, we propose a novel analytic transfer matrix method (ATMM) to the analysis of planar waveguides with index discontinuity or index slope discontinuity, for the cases when the conventional WKB method is no longer valid. We also analyze the physical insight of the approximations in WKB theory, and according to our research, the phase shift at the turning point is not /2, but exactly . Test calculations are done for an index profile with a known solution and the comparison shows that our method gives extremely accurate propagation constant.     

Strongly Regular Graphs and Designs with Three Intersection Numbers

We investigate BIBDs with three intersection numbers,x, y, and z, such that the relation on the block set given byA B iff the cardinality of the intersectionof A and B is not equal to x is an equivalence relation. Withsuch a design, we associate a family of strongly regular graphswith the same parameters. Two constructions producing infinitefamilies of such designs are given.