Large Cliques in -Free Graphs

Dedicated to the memory of Paul Erdős A graph is called -free if it contains no cycle of length four as an induced subgraph. We prove that if a -free graph has n vertices and at least edges then it has a complete subgraph of vertices, where depends only on . We also give estimates on and show that a similar result does not hold for H-free graphs––unless H is an induced subgraph of . The best value of is determined for chordal graphs. Received October 25, 1999 RID="*" ID="*" Supported by OTKA grant T029074. RID="**" ID="**" Supported by TKI grant stochastics@TUB and by OTKA grant T026203.     

An augmenting path algorithm for linear matroid parity

Linear matroid parity generalizes matroid intersection and graph matching (and hence network flow, degree-constrained subgraphs, etc.). A polynomial algorithm was given by Lovász. This paper presents an algorithm that uses timeO(mn ³), wherem is the number of elements andn is the rank. (The time isO(mn ^2.5) using fast matrix multiplication; both bounds assume the uniform cost model). For graphic matroids the time isO(mn ²). The algorithm is based on the method of augmenting paths used in the algorithms for all subcases of the problem. First author was supported in part by the National Science Foundation under grants MCS 78-18909, MCS-8302648, and DCR-8511991. The research was done while the second author was at the University of Denver and at the University of Colorado at Boulder.     

Nonexistence of universal graphs without some trees

IfG is a finite tree with a unique vertex of largest, and 4 degree which is adjacent to a leaf then there is no universal countableG-free graph.Research partially supported by the Hungarian Science Research Grant OTKA No. 2117 and by the European Communities (Cooperation in Science and Technology with Central and Eastern European Countries) contract number ERBCIPACT930113.     

Perfect couples of graphs

We generalize the concept of perfect graphs in terms of additivity of a functional called graph entropy. The latter is an information theoretic functional on a graphG with a probability distributionP on its vertex set. For any fixedP it is sub-additive with respect to graph union. The entropy of the complete graph equals the sum of those ofG and its complement G iffG is perfect. We generalize this recent result to characterize all the cases when the sub-additivity of graph entropy holds with equality.The research of the authors is partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant No. 1806 resp. No. 1812.     

Graph Orientations with Edge-connection and Parity Constraints

Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset with |E| + |T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k)-edge-connected graph with |V| + |E| even has a (k-1)-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k edge-disjoint paths from s to every node and k-1 edge-disjoint paths from every node to s. Received December 14, 1998/Revised January 12, 2001 RID="*" ID="*" Supported by the Hungarian National Foundation for Scientific Research, OTKA T029772. Part of research was done while this author was visiting EPFL, Lausanne, June, 1998. RID="†" ID="†" Supported by the Hungarian National Foundation for Scientific Research, OTKA T029772 and OTKA T030059.     

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K_{⌈t/2⌉,⌊t/2⌋} in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme. * Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046376, AT048826, and NK62321. † Research partially supported by the NSERC grant 611470 and the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046234, AT048826, and NK62321.     

Symplectic Spaces And Ear-Decomposition Of Matroids

Matroids admitting an odd ear-decomposition can be viewed as natural generalizations of factor-critical graphs. We prove that a matroid representable over a field of characteristic 2 admits an odd ear-decomposition if and only if it can be represented by some space on which the induced scalar product is a non-degenerate symplectic form. We also show that, for a matroid representable over a field of characteristic 2, the independent sets whose contraction admits an odd ear-decomposition form the family of feasible sets of a representable Δ-matroid.     

Recognizing graphic matroids

There is no polynomially bounded algorithm to test if a matroid (presented by an “independence oracle”) is binary. However, there is one to test graphicness. Finding this extends work of previous authors, who have given algorithms to test binary matroids for graphicness. Our main tool is a new result that ifM′ is the polygon matroid of a graphG, andM is a different matroid onE(G) with the same rank, then there is a vertex ofG whose star is not a cocircuit ofM.     

Polynomial Time Recognition of Uniform Cocircuit Graphs

We present an algorithm which takes a graph as input and decides in polynomial time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid.     

Equivalent factor matroids of graphs

The factor matroid of a graphG is the matric matroid of the vertex-edge incidence matrix ofG interpreted over the real numbers. This paper presents a constructive characterization of the graphs hat have the same factor matroid as a given 4-connected bipartite graph.Research partially supported by NSF Grant ESS-8307796 and Office of Naval Research Grant N00014-86-K-0689.     

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.     

Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs

Recently much attention has been focused on the theory of quasi-random graph and hypergraph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. We shall investigate propertiesP which do not imply quasi-randomnes for sequences (G _n ) of graphs on their own, but do imply if they hold not only for the whole graphG _n but also for every sufficiently large subgraph ofG _n . Here the properties are strongly connected to countingnot necessarily induced subgraphs of a given type, while in a subsequent paper we shall investigate the properties connected with counting induced subgraphs.Dedicated to the memory of Paul ErdsResearch supported by OTKA N1909.     

The inertia of unicyclic graphs and the implications for closed-shells

The inertia of a graph is an integer triple specifying the number of negative, zero, and positive eigenvalues of the adjacency matrix of the graph. A unicyclic graph is a simple connected graph with an equal number of vertices and edges. This paper characterizes the inertia of a unicyclic graph in terms of maximum matchings and gives a linear-time algorithm for computing it. Chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is (properly) closed-shell, having exactly half of its eigenvalues greater than zero, because this designates a stable electron configuration. The inertia determines whether a graph is closed-shell, and hence the reported result gives a linear-time algorithm for determining this for unicyclic graphs.     

Coordinatizing R-trees in terms of universal c-trees

A complete ℝ-treeT will be constructed such that, for everyxσT, the cardinality of the set of connected components ofT{x} is the same and equals a pre-given cardinalityc; by this construction simultaneously the valuated matroid of the ends of this ℝ-tree is given. In addition, for any arbitrary ℝ-tree, an embedding into such a “universalc-tree” (for suitablec) will be constructed.     

Packing Non-Returning <Emphasis Type="Italic">A</Emphasis>-Paths*

Chudnovsky et al. gave a min-max formula for the maximum number of node-disjoint nonzero A-paths in group-labeled graphs [1], which is a generalization of Mader's theorem on node-disjoint A-paths [3]. Here we present a further generalization with a shorter proof. The main feature of Theorem 2.1 is that parity is “hidden” inside , which is given by an oracle for non-bipartite matching. * Research is supported by OTKA grants T 037547 and TS 049788, by European MCRTN Adonet, Contract Grant No. 504438 and by the Egerváry Research Group of the Hungarian Academy of Sciences. † The author is a member of the Egerváry Research Group (EGRES).     

The ellipsoid method and its consequences in combinatorial optimization

L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields a proof that weighted fractional chromatic number is NP-hard. Research by the third author was supported by the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.).     

Total Colorings Of Degenerate Graphs

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is s-degenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤s. We prove that an s-degenerate graph G has a total coloring with Δ+1 colors if the maximum degree Δ of G is sufficiently large, say Δ≥4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a lineartime algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.