Pathwidth of Planar and Line Graphs

 We prove that for every 2-connected planar graph the pathwidth of its geometric dual is less than the pathwidth of its line graph. This implies that pathwidth(H)≤ pathwidth(H ^*)+1 for every planar triangulation H and leads us to a conjecture that pathwidth(G)≤pathwidth(G ^*)+1 for every 2-connected graph G. Received: May 8, 2001 Final version received: March 26, 2002 RID="*" ID="*" I acknowledge support by EC contract IST-1999-14186, Project ALCOM-FT (Algorithms and Complexity - Future Technologies) and support by the RFBR grant N01-01-00235. Acknowledgments. I am grateful to Petr Golovach, Roland Opfer and anonymous referee for their useful comments and suggestions.     

Locally Well-Dominated and Locally Independent Well-Dominated Graphs

 In this article we present characterizations of locally well-dominated graphs and locally independent well-dominated graphs, and a sufficient condition for a graph to be k-locally independent well-dominated. Using these results we show that the irredundance number, the domination number and the independent domination number can be computed in polynomial time within several classes of graphs, e.g., the class of locally well-dominated graphs. Received: September 13, 2001 Final version received: May 17, 2002 RID="*" ID="*" Supported by the INTAS and the Belarus Government (Project INTAS-BELARUS 97-0093) RID="†" ID="†" Supported by RUTCOR RID="*" ID="*" Supported by the INTAS and the Belarus Government (Project INTAS-BELARUS 97-0093) 05C75, 05C69 Acknowledgments. The authors thank the referees for valuable suggestions.     

Maximal Energy Bipartite Graphs

 Given a graph G, its energy E(G) is defined to be the sum of the absolute values of the eigenvalues of G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. Here we show that if G is a bipartite graph with n vertices, then

The Hilbert Series of the Face Ring of a Flag Complex

 It is shown that the Hilbert series of the face ring of a clique complex (equivalently, flag complex) of a graph G is, up to a factor, just a specialization of , the subgraph polynomial of the complement of G. We also find a simple relationship between the size of a minimum vertex cover of a graph G and its subgraph polynomial. This yields a formula for the h-vector of the flag complex in terms of those two invariants of . Some computational issues are addressed and a recursive formula for the Hilbert series is given based on an algorithm of Bayer and Stillman. Received: December 10, 1999 Acknowledgments. I would like to thank Rick Wilson and the mathematics department of the California Institute of Technology for their kind hospitality, and Richard Stanley for pointing out an error in an earlier draft.     

Intersecting Curves in the Plane

 We prove that for every family of n pairwise intersecting simple closed planar curves in general position, at least (4/5)n ²−O(n) points lie on more than one curve. This improves the previous lower bound of (3/4)n ²−O(n) due to Richter and Thomassen. Received: March 29, 2000 Final version received: August 30, 2001 RID="*" ID="*" Research supported in part by NSF grant DMS-9970325 Acknowledgments. I thank Bruce Richter for informing me about this problem, Gelasio Salazar for reading a preliminary version of the paper, and a Referee for useful comments. Current Address: Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA. e-mail: mubayi@microsoft.com 1991 Mathematics Subject Classification. 05C35, 52C10     

Vertex Covers by Edge Disjoint Cliques

Dedicated to the memory of Paul Erdős Let H be a simple graph having no isolated vertices. An (H,k)-vertex-cover of a simple graph G = (V,E) is a collection of subgraphs of G satisfying 1.  , for all i = 1, ..., r, 2.  , 3.  , for all , and 4.  each is in at most k of the . We consider the existence of such vertex covers when H is a complete graph, , in the context of extremal and random graphs. Received October 31, 1999 RID="*" ID="*" Supported in part by NSF grant DMS-9627408. RID="†" ID="†" Supported in part by NSF grant CCR-9530974. RID="‡" ID="‡" Supported in part by OTKA Grants T 030059 and T 29074, FKFP 0607/1999 and by the Bolyai Foundation. RID="§" ID="§" Supported in part by NSF grant DMS-9970622.     

The Clique Numbers of Regular Graphs

 Let ω(G) be the clique number of a graph G. We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values ω(G) completely cover a line segment [a,b] of positive integers. For an arbitrary graphic degree sequence d, we define min(ω,d) and max(ω,d) as follows:

On the Oka principle in a Banach space,I

 Let X be a complex Banach space with a countable unconditional basis, Ω⊂X pseudoconvex open, G a complex Banach Lie group. We show that a Runge–type approximation hypothesis on X, G (which we also prove for G a solvable Lie group) implies that any holomorphic cocycle on Ω with values in G can be resolved holomorphically if it can be resolved continuously. Received: 1 March 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 32L05, 32E30, 46G20 RID="*" ID="*" Kedves Szímuskának. RID="*" ID="*" To my dear Wife.     

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     

Fusion Relation in Products of Association Schemes

 Fusion relations between the association schemes obtained by direct product and wreath product are established via a study of their matrix representations. The character table of the scheme obtained by the wreath product is described and some algebraic properties of the products are derived. Received: May 7, 1999 Final version received: September 24, 1999 RID="*" ID="*" 1991 Mathematics Subject Classification. Primary 05E30; Secondary 05B99 RID="*" ID="*" Supported in part by Com ²MaC-KOSEF, POSTECH, Korea Acknowledgments. The author is indebted to an anonymous referee who provided the complete proof of Theorem 4.2.     

On the structure of rotation-invariant semigroups

 We generalize the notions of Girard algebras and MV-algebras by introducing rotation-invariant semigroups. Based on a geometrical characterization, we present five construction methods which result in rotation-invariant semigroups and in particular, Girard algebras and MV-algebras. We characterize divisibility of MV-algebras, and point out that integrality of Girard algebras follows from their other axioms. Received: 7 January 2002 / Revised version: 4 April 2002 / Published online: 19 December 2002 RID="*" ID="*" Supported by the National Scientific Research Fund Hungary (OTKA F/032782). Mathematics Subject Classification (2000): 20M14, 06F05 Key words or phrases: Residuated lattice – Conjunction for non-classical logics     

Uniform Group Divisible Designs with Block Sizes Three and <Emphasis Type="Italic">n</Emphasis>

 The existence of group divisible designs of type g ^t with block sizes three and n, 4≤ n≤10, is completely settled for all values of g and t. Received: July 21, 1999 Final version received: September 10, 2001 RID="*" ID="*" This work was done in 1995 while the authors were graduate students at the University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Acknowledgments. The authors would like to thank the referee for his careful reading and for pointing out some errors in an earlier draft of the paper.     

On the Caccetta–Häggkvist Conjecture

 It was conjectured by Caccetta and H?ggkvist in 1978 that every digraph G with n vertices and minimum outdegree at least r contains a directed cycle of length at most ⌈n/r⌉. By refining an argument of Chvátal and Szemerédi, we prove that such G contains a directed cycle of length at most n/r+73. Received: September 13, 1999 Final version received: June 19, 2000 Acknowledgment. I want to thank a referee for many valuable suggestions leading to the clear presentation of the paper.     

Choosability of Powers of Circuits

 It is proved that ch(G)=χ(G) if G=C _n ^p , the pth power of the circuit graph C _n , or if G is a uniform inflation of such a graph. The proof uses the method of Alon and Tarsi. As a corollary, the (a : b)-choosability conjectures hold for all such graphs. Received: October 10, 2000 Final version received: November 8, 2001     

On Ramsey Numbers for Trees Versus Wheels of Five or Six Vertices

 For given two graphs G dan H, the Ramsey number R(G,H) is the smallest positive integer n such that every graph F of order n must contain G or the complement of F must contain H. In [12], the Ramsey numbers for the combination between a star S _n and a wheel W _m for m=4,5 were shown, namely, R(S _n ,W ₄)=2n−1 for odd n and n≥3, otherwise R(S _n ,W ₄)=2n+1, and R(S _n ,W ₅)=3n−2 for n≥3. In this paper, we shall study the Ramsey number R(G,W _m ) for G any tree T _n . We show that if T _n is not a star then the Ramsey number R(T _n ,W ₄)=2n−1 for n≥4 and R(T _n ,W ₅)=3n−2 for n≥3. We also list some open problems. Received: October, 2001 Final version received: July 11, 2002 RID="*" ID="*" This work was supported by the QUE Project, Department of Mathematics ITB Indonesia Acknowledgments. We would like to thank the referees for several helpful comments.     

The Convexity Number of a Graph

 For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a u−v shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,v∈S. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented. Received: August 19, 1998 Final version received: May 17, 2000     

Graphs of Non-Crossing Perfect Matchings

 Let P _n be a set of n=2m points that are the vertices of a convex polygon, and let ℳ _m be the graph having as vertices all the perfect matchings in the point set P _n whose edges are straight line segments and do not cross, and edges joining two perfect matchings M ₁ and M ₂ if M ₂=M ₁−(a,b)−(c,d)+(a,d)+(b,c) for some points a,b,c,d of P _n . We prove the following results about ℳ _m : its diameter is m−1; it is bipartite for every m; the connectivity is equal to m−1; it has no Hamilton path for m odd, m>3; and finally it has a Hamilton cycle for every m even, m≥4. Received: October 10, 2000 Final version received: January 17, 2002 RID="*" ID="*" Partially supported by Proyecto DGES-MEC-PB98-0933 Acknowledgments. We are grateful to the referees for comments that helped to improve the presentation of the paper.     

Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also "randomizing" given graphs. The main specific result that we prove here (Theorem 1) is that if is the smallest vertex degree in G, then almost all n-covers of G are -connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. Received June 21, 1999/Revised November 16, 2000 RID="*" ID="*" Work supported in part by grants from the Israel Academy of Aciences and the Binational Israel-US Science Foundation.     

A Contribution to a Conjecture of A. Saito

 A. Saito conjectured that every finite 3-connected line graph of diameter at most 3 is hamiltonian unless it is the line graph of a graph obtained from the Petersen graph by adding at least one pendant edge to each of its vertices. Here we shall see that a line graph of connectivity 3 and diameter at most 3 has a hamiltonian path. Received: May 31, 2000 Final version received: August 17, 2001 RID="*" ID="*" This work has partially been supported by DIMATIA, Grant 201/99/0242, Grant Agency of the Czech Republic AMS subject classification: 05C45, 05C40