The Mysterious 2-Crown

We show that the 2-crown is not coproductive, which is to say that the class of those bounded distributive lattices whose Priestley spaces lack any copy of the 2-crown is not productive. We do this by first exhibiting a general construction to handle questions of this sort. We then use a particular instance of this constrution, along with some of the combinatorial features of projective planes, to show that the 2-crown is not coproductive. This paper is dedicated to Walter Taylor. Received November 24, 2004; accepted in final form July 16, 2005. The first author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic. The second author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic, by the NSERC of Canada and by the Gudder Trust of the University of Denver. The third author would like to express his thanks for support by the NSERC of Canada and partial support by project LN 00A056 of the Ministry of Education of the Czech Republic.     

On the Pancyclicity of Lexicographic Products

We prove that if G and H are graphs containing at least one edge each, then their lexicographic product G[H] is weakly pancyclic, i. e., it contains a cycle of every length between the length of a shortest cycle and that of a longest one. This supports some conjectures on locally connected graphs and on product graphs. We obtain an analogous result on even cycles in products G[H] that are bipartite. We also investigate toughness conditions on G implying that G[H] is hamiltonian (and hence pancyclic). Supported by the project LN00A056 of the Czech Ministry of Education     

Equipartite graphs

A graph G of even order is weakly equipartite if for any partition of its vertex set into subsets V ₁ and V ₂ of equal size the induced subgraphs G[V ₁] and G[V ₂] are isomorphic. A complete characterization of (weakly) equipartite graphs is derived. In particular, we show that each such graph is vertex-transitive. In a subsequent paper, we use these results to characterize equipartite polytopes, a geometric analogue of equipartite graphs. Supported by Research Plan MSM 4977751301 of the Czech Ministry of Education. The Institute for Theoretical Computer Science is supported by Czech Ministry of Education as projects LN00A056 and 1M0545. Support by the Fulbright Senior Specialist Program for his stay in Pilsen in summer 2003 is acknowleged.     

Short Disjoint Paths in Locally Connected Graphs

Chartrand and Pippert proved that a connected, locally k-connected graph is (k + 1)-connected. We investigate the lengths of k + 1 disjoint paths between two vertices in locally k-connected graphs with respect to several graph parameters, e.g. the k-diameter of a graph. We also give a generalization of the mentioned result. This research was partly done on a visit to the Institute of Systems Science of Academia Sinica (Beijing, China) under the project ME 418 of the Czech Ministery of Education. These authors are partly supported by the project LN00A056 of the Czech Ministery of Education.     

A Priestley Sum of Finite Trees is Acyclic

We show that the Priestley sum of finite trees contains no cyclic finite poset. The first author would like to express his thanks for support from project LN 1M0021620808 of the Ministry of Education of the Czech Republic. The second author would like to express his thanks for support from project 1M0021620808 of the Ministry of Education of the Czech Republic, from the NSERC of Canada and from a PROF grant from the University of Denver. The third author would like to express his thanks for the support from the NSERC of Canada and for partial support from the project 1M0021620808 of the Ministry of Education of the Czech Republic.     

Random Walks And The Colored Jones Function

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones function: one in terms of the permanent of a matrix associated to a chord diagram, and another in terms of counting paths of intersecting chords. Electronic supplementary material to this article is available at and is accessible to authorized users. * S. G. was partially supported by an NSF and by an Israel-US BSF grant. † M. L. was partly supported by GAUK 158 grant and by the Project LN00A056 of the Czech Ministry of Education.     

A Note on the Shortness Coefficient and the Hamiltonicity of 4-Connected Line Graphs

Thomassen proposed a well-known conjecture: every 4-connected line graph is hamiltonian. In this note, we show that Thomassens conjecture is equivalent to the statement that the shortness coefficient of the class of all 4-connected line graphs is one and the statement that the shortness coefficient of the class of all 4-connected claw-free graphs is one respectively.Research partially supported by the fund of the basic research of Beijing Institute of Technology, by the fund of Natural Science of Jiangxi Province and by grant No. LN00A056 of the Czech Ministry of Education     

Finite Paths are Universal

We prove that any countable (finite or infinite) partially ordered set may be represented by finite oriented paths ordered by the existence of homomorphism between them. This (what we believe a surprising result) solves several open problems. Such path-representations were previously known only for finite and infinite partial orders of dimension 2. Path-representation implies the universality of other classes of graphs (such as connected cubic planar graphs). It also implies that finite partially ordered sets are on-line representable by paths and their homomorphisms. This leads to a new on-line dimensions. *Supported by Grants LN00A56 and 1M0021620808 of the Czech Ministry of Education. The first author was partially supported by EU network COMBSTRU at UPC Barcelona.     

Coloring even-faced graphs in the torus and the Klein bottle

We prove that a triangle-free graph drawn in the torus with all faces bounded by even walks is 3-colorable if and only if it has no subgraph isomorphic to the Cayley graph C(Z ₁₃; 1,5). We also prove that a non-bipartite quadrangulation of the Klein bottle is 3-colorable if and only if it has no non-contractible separating cycle of length at most four and no odd walk homotopic to a non-contractible two-sided simple closed curve. These results settle a conjecture of Thomassen and two conjectures of Archdeacon, Hutchinson, Nakamoto, Negami and Ota. Institute for Theoretical Computer Science is supported as project 1M0545 by the Ministry of Education of the Czech Republic. The author was visiting Georgia Institute of Technology as a Fulbright scholar in the academic year 2005/06. Partially supported by NSF Grants No. DMS-0200595 and DMS-0354742.     

Finite Paths are Universal

We prove that any countable (finite or infinite) partially ordered set may be represented by finite oriented paths ordered by the existence of homomorphism between them. This (what we believe a surprising result) solves several open problems. Such path-representations were previously known only for finite and infinite partial orders of dimension 2. Path-representation implies the universality of other classes of graphs (such as connected cubic planar graphs). It also implies that finite partially ordered sets are on-line representable by paths and their homomorphisms. This leads to new on-line dimensions. Mathematics Subject Classifications (2000) 06A06, 06A07, 05E99, 05C99.J. Nešetřil: Supported by a Grant LN00A56 of the Czech Ministry of Education. The first author was partially supported by EU network COMBSTRU at UPC Barcelona.     

List Homomorphisms and Circular Arc Graphs

G , H, and lists , a list homomorphism of G to H with respect to the lists L is a mapping , such that for all , and for all . The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists , , admits a list homomorphism with respect to L. We have introduced the list homomorphism problem in an earlier paper, and proved there that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP-complete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NP-complete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallai's asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two. Received: July 22, 1996/Revised: Revised June 10, 1998     

Colorings Of Plane Graphs With No Rainbow Faces

We prove that each n-vertex plane graph with girth g≥4 admits a vertex coloring with at least ⌈n/2⌉+1 colors with no rainbow face, i.e., a face in which all vertices receive distinct colors. This proves a conjecture of Ramamurthi and West. Moreover, we prove for plane graph with girth g≥5 that there is a vertex coloring with at least if g is odd and if g is even. The bounds are tight for all pairs of n and g with g≥4 and n≥5g/2−3. * Supported in part by the Ministry of Science and Technology of Slovenia, Research Project Z1-3129 and by a postdoctoral fellowship of PIMS. ** Institute for Theoretical Computer Science is supported by Ministry of Education of CzechR epublic as project LN00A056.     

Long paths in sparse random graphs

We consider random graphs withn labelled vertices in which edges are chosen independently and with probabilityc/n. We prove that almost every random graph of this kind contains a path of length ≧(1 −α(c))n where α(c) is an exponentially decreasing function ofc. Dedicated to Tibor Gallai on his seventieth birthday     

Compactness results in extremal graph theory

Let L be a given family of so called prohibited graphs. Let ex (n, L) denote the maximum number of edges a simple graph of ordern can have without containing subgraphs from L. A typical extremal graph problem is to determine ex (n, L), or at least, find good bounds on it. Results asserting that for a given L there exists a much smaller L*⫅L for which ex (n, L) ≈ ex (n, L*) will be calledcompactness results. The main purpose of this paper is to prove some compactness results for the case when L consists of cycles. One of our main tools will be finding lower bounds on the number of pathsP ^k+1 in a graph ofn vertices andE edges., witch is, in fact, a “super-saturated” version of a wellknown theorem of Erdős and Gallai. Dedicated to Tibor Gallai on his seventieth birthday     

Diperfect graphs

Gallai and Milgram have shown that the vertices of a directed graph, with stability number α(G), can be covered by exactly α(G) disjoint paths. However, the various proofs of this result do not imply the existence of a maximum stable setS and of a partition of the vertex-set into paths μ₁, μ₂, ..., μ_k such tht |μ_i ∩S|=1 for alli. Later, Gallai proved that in a directed graph, the maximum number of vertices in a path is at least equal to the chromatic number; here again, we do not know if there exists an optimal coloring (S ₁,S ₂, ...,S _k) and a path μ such that |μ ∩S _i|=1 for alli. In this paper we show that many directed graphs, like the perfect graphs, have stronger properties: for every maximal stable setS there exists a partition of the vertex set into paths which meet the stable set in only one point. Also: for every optimal coloring there exists a path which meets each color class in only one point. This suggests several conjecties similar to the perfect graph conjecture. Dedicated to Tibor Gallai on his seventieth birthday     

Hajós Theorem For Colorings Of Edge-Weighted Graphs

Hajós theorem states that every graph with chromatic number at least k can be obtained from the complete graph K _k by a sequence of simple operations such that every intermediate graph also has chromatic number at least k. Here, Hajós theorem is extended in three slightly different ways to colorings and circular colorings of edge-weighted graphs. These extensions shed some new light on the Hajós theorem and show that colorings of edge-weighted graphs are most natural extension of usual graph colorings.* Supported in part by the Ministry of Education, Science and Sport of Slovenia, Research Program P0–0507–0101.     

Some remarks on interval graphs

Using simplicial decompositions a new and simple proof of Lekkerkerker-Boland’s criterion for interval graphs is given. Also the infinite case is considered, and the problem is tackled to what extent the representation of a graph as an interval graph is unique. Dedicated to Tibor Gallai on his seventieth birthday     

Davenport–Schinzel Trees*

In this paper we study labeled–tree analogues of (generalized) Davenport–Schinzel sequences.We say that two sequences a ₁ ... a _k , b ₁ ... b _k of equal length k are isomorphic, if a _i = a _j i b _i = b _j (for all i, j). For example, the sequences 11232, 33141 are isomorphic. We investigate the maximum size of a labeled (rooted) tree with each vertex labeled by one of n labels in such a way that, besides some technical conditions, the sequence of labels along any path (starting from the root) contains no subsequence isomorphic to a fixed forbidden sequence u.We study two models of such labeled trees. Each of the models is known to be essentially equivalent also to other models. The labeled paths in a special case of one of our models correspond to classical Davenport–Schinzel sequences.We investigate, in particular, for which sequences u the labeled tree has at most O(n) vertices. In both models, we answer this question for any forbidden sequence u over a two-element alphabet and also for a large class of other sequences u.* This research was partially supported by Charles University grants No. 99/158 and 99/159 and by Czech Republic Grant GAR 201/99/0242. Supported by project LN00A056 of The Ministry of Education of the Czech Republic.     

Ramsey‐type results for Gallai colorings

A Gallai‐coloring of a complete graph is an edge coloring such that no triangle is colored with three distinct colors. Gallai‐colorings occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper) or information theory. Gallai‐colorings extend 2‐colorings of the edges of complete graphs. They actually turn out to be close to 2‐colorings—without being trivial extensions. Here, we give a method to extend some results on 2‐colorings to Gallai‐colorings, among them known and new, easy and difficult results. The method works for Gallai‐extendible families that include, for example, double stars and graphs of diameter at most d for 2?d, or complete bipartite graphs. It follows that every Gallai‐colored K_n contains a monochromatic double star with at least 3n+ 1/4 vertices, a monochromatic complete bipartite graph on at least n/2 vertices, monochromatic subgraphs of diameter two with at least 3n/4 vertices, etc. The generalizations are not automatic though, for instance, a Gallai‐colored complete graph does not necessarily contain a monochromatic star on n/2 vertices. It turns out that the extension is possible for graph classes closed under a simple operation called equalization. We also investigate Ramsey numbers of graphs in Gallai‐colorings with a given number of colors. For any graph H let RG(r, H) be the minimum m such that in every Gallai‐coloring of K_m with r colors, there is a monochromatic copy of H. We show that for fixed H, RG (r, H) is exponential in r if H is not bipartite; linear in r if H is bipartite but not a star; constant (does not depend on r) if H is a star (and we determine its value). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 233–243, 2010     

Graph congruences and what they connote

Abstract

The algebraic notion of a “congruence” seems to be foreign to contemporary graph theory. We propound that it need not be so by developing a theory of congruences of graphs: a congruence on a graph G = (V, E) being a pair (～, ) of which ～ is an equivalence relation on V and is a set of unordered pairs of vertices of G with a special relationship to ～ and E. Kernels and quotient structures are used in this theory to develop homomorphism and isomorphism theorems which remind one of similar results in an algebraic context. We show that this theory can be applied to deliver structural decompositions of graphs into “factor” graphs having very special properties, such as the result that each graph, except one, is a subdirect product of graphs with universal vertices. In a final section, we discuss corresponding concepts and briefly describe a corresponding theory for graphs which have a loop at every vertex and which we call loopy graphs. They are in a sense more “algebraic” than simple graphs, with their meet-semilattices of all congruences becoming complete algebraic lattices.