Spanning triangulations in graphs

We prove that every graph of sufficiently large order n and minimum degree at least 2n/3 contains a triangulation as a spanning subgraph. This is best possible: for all integers n, there are graphs of order n and minimum degree ?2n/3? ? 1 without a spanning triangulation. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Dense graphs with small clique number

We consider the structure of K_r‐free graphs with large minimum degree, and show that such graphs with minimum degree δ>(2r ? 5)n/(2r ? 3) are homomorphic to the join K_{r ? 3}∨H, where H is a triangle‐free graph. In particular this allows us to generalize results from triangle‐free graphs and show that K_r‐free graphs with such a minimum degree have chromatic number at most r +1. We also consider the minimum‐degree thresholds for related properties. Copyright © 2010 John Wiley & Sons, Ltd. 66:319‐331, 2011     

Oriented walk double covering and bidirectional double tracing

An oriented walk double covering of a graph G is a set of oriented closed walks, that, traversed successively, combined will have traced each edge of G once in each direction. A bidirectional double tracing of a graph G is an oriented walk double covering that consists of a single closed walk. A retracting in a closed walk is the immediate succession of an edge by its inverse. Every graph with minimum degree 2 has a retracting free oriented walk double covering and every connected graph has a bidirectional double tracing. The minimum number of closed walks in a retracting free oriented walk double covering of G is denoted by c(G). The minimum number of retractings in a bidirectional double tracing of G is denoted by r(G). We shall prove that for all connected noncycle graphs G with minimum degree at least 2, r(G) = c(G) − 1. The problem of characterizing those graphs G for which r(G) = 0 was raised by Ore. Thomassen solved this problem by relating it to the existence of certain spanning trees. We generalize this result, and relate the parameters r(G), c(G) to spanning trees of G. This relation yields a polynomial time algorithm to determine the parameters c(G) and r(G). © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 89–102, 1998     

Almost all regular graphs are hamiltonian

In a previous article the authors showed that almost all labelled cubic graphs are hamiltonian. In the present article, this result is used to show that almost all r-regular graphs are hamiltonian for any fixed r ? 3, by an analysis of the distribution of 1-factors in random regular graphs. Moreover, almost all such graphs are r-edge-colorable if they have an even number of vertices. Similarly, almost all r-regular bipartite graphs are hamiltonian and r-edge-colorable for fixed r ? 3. © 1994 John Wiley & Sons, Inc.     

Infinite families of crossing‐critical graphs with prescribed average degree and crossing number

?iráň constructed infinite families of k‐crossing‐critical graphs for every k?3 and Kochol constructed such families of simple graphs for every k?2. Richter and Thomassen argued that, for any given k?1 and r?6, there are only finitely many simple k‐crossing‐critical graphs with minimum degree r. Salazar observed that the same argument implies such a conclusion for simple k‐crossing‐critical graphs of prescribed average degree r>6. He established the existence of infinite families of simple k‐crossing‐critical graphs with any prescribed rational average degree r∈[4, 6) for infinitely many k and asked about their existence for r∈(3, 4). The question was partially settled by Pinontoan and Richter, who answered it positively for $r\in(3\frac{1}{2},4)$. The present contribution uses two new constructions of crossing‐critical simple graphs along with the one developed by Pinontoan and Richter to unify these results and to answer Salazar's question by the following statement: there exist infinite families of simple k‐crossing‐critical graphs with any prescribed average degree r∈(3, 6), for any k greater than some lower bound N_r. Moreover, a universal lower bound N_I on k applies for rational numbers in any closed interval I?(3, 6). © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 139–162, 2010     

A Network-Flow Technique for Finding Low-Weight Bounded-Degree Spanning Trees

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low-weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning treeTusingadoptionsto meet the degree constraints is considered. A novel network-flow-based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previous algorithm. If the degree constraintd(v) for eachvis at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 − min{(d(v) − 2)/(deg_T(v) − 2) : deg_T(v) > 2}, where deg_T(v) is the initial degree ofv. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds foranyalgorithm at all, then it also holds for the given algorithm. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. ChoosingTto be a minimum spanning tree yields approximation algorithms with factors less than 2 for the general problem on geometric graphs with distances induced by variousL_pnorms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.     

Matching signatures and Pfaffian graphs

Perfect matchings of k-Pfaffian graphs may be enumerated in polynomial time on the number of vertices, for fixed k. In general, this enumeration problem is #P-complete. We give a Composition Theorem of 2r-Pfaffian graphs from r Pfaffian spanning subgraphs. Constructions of k-Pfaffian graphs known prior to this seem to be of a very different and essentially topological nature. We apply our Composition Theorem to produce a bipartite graph on 10 vertices that is 6-Pfaffian but not 4-Pfaffian. This is a counter-example to a conjecture of Norine (2009) [8], which states that the Pfaffian number of a graph is a power of four.     

Rainbow matchings in Dirac bipartite graphs

We show the existence of rainbow perfect matchings in μn‐bounded edge colorings of Dirac bipartite graphs, for a sufficiently small μ > 0. As an application of our results, we obtain several results on the existence of rainbow k‐factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree.     

Crossing numbers of sequences of graphs II: Planar tiles

We describe a method of creating an infinite family of crossing‐critical graphs from a single small planar map, the tile, by gluing together many copies of the tile together in a circular fashion. This method yields all known infinite families of k‐crossing‐critical graphs. Furthermore, the method yields new infinite families, which extend from (4,6) to (3.5,6) the interval of rationals r for which there is, for some k, an infinite sequence of k‐crossing‐critical graphs all having average degree r. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 332–341, 2003     

A Necessary and Sufficient Condition for the Existence of a Heterochromatic Spanning Tree in a Graph

We prove the following theorem. An edge-colored (not necessary to be proper) connected graph G of order n has a heterochromatic spanning tree if and only if for any r colors (1≤r≤n−2), the removal of all the edges colored with these r colors from G results in a graph having at most r+1 components, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors.     

Spanning Star Trees in Regular Graphs

For a subset W of vertices of an undirected graph G, let S(W) be the subgraph consisting of W, all edges incident to at least one vertex in W, and all vertices adjacent to at least one vertex in W. If S(W) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that for every r ≥ 3, there exist r-regular n-vertex graphs that have spanning star trees, and there exist r-regular n-vertex graphs that do not have spanning star trees, for all n sufficiently large (in terms of r). Furthermore, the problem of determining whether a given regular graph has a spanning star tree is NP-complete.     

Branchings in rooted graphs and the diameter of greedoids

LetG be a 2-connected rooted graph of rankr andA, B two (rooted) spanning trees ofG We show that the maximum number of exchanges of leaves that can be required to transformA intoB isr ²–r+1 (r>0). This answers a question by L. Lovász.There is a natural reformulation of this problem in the theory ofgreedoids, which asks for the maximum diameter of the basis graph of a 2-connected branching greedcid of rankr.Greedoids are finite accessible set systems satisfying the matroid exchange axiom. Their theory provides both language and conceptual framework for the proof. However, it is shown that for general 2-connected greedoids (not necessarily constructed from branchings in rooted graphs) the maximum diameter is 2^r–1.     

A GRASP algorithm for the multi-criteria minimum spanning tree problem

This paper proposes a GRASP (Greedy Randomized Adaptive Search Procedure) algorithm for the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem a vector of costs is defined for each edge of the graph and the problem is to find all Pareto optimal or efficient spanning trees (solutions). The algorithm is based on the optimization of different weighted utility functions. In each iteration, a weight vector is defined and a solution is built using a greedy randomized constructive procedure. The found solution is submitted to a local search trying to improve the value of the weighted utility function. We use a drop-and-add neighborhood where the spanning trees are represented by Prufer numbers. In order to find a variety of efficient solutions, we use different weight vectors, which are distributed uniformly on the Pareto frontier. The proposed algorithm is tested on problems with r=2 and 3 criteria. For non-complete graphs with n=10, 20 and 30 nodes, the performance of the algorithm is tested against a complete enumeration. For complete graphs with n=20, 30 and 50 nodes the performance of the algorithm is tested using two types of weighted utility functions. The algorithm is also compared with the multi-criteria version of the Kruskal’s algorithm, which generates supported efficient solutions. This work was funded by the Municipal Town Hall of Campos dos Goytacazes city. The used computer was acquired with resource of CNPq.     

Regular planar graphs with faces of only two types and shortness parameters

We construct a small non-Hamiltonian 3-connected trivalent planar graph whose faces are all 4-gons or 7-gons and show that the shortness coefficient of the class of such graphs is less than one. Then, by transforming non-Hamiltonian trivalent graphs into regular graphs of valency four or five, we obtain our main results, as follows. We show first that the class of 3-connected r-valent planar graphs whose faces are of only two types, triangles and q-gons, contains non-Hamiltonian members and has a shortness exponent less than one when r = 4, for all q ≧ 12. Under the extra restriction that, among graphs of connectivity three, only those with maximum cyclic edge-connectivity are to be considered, we prove the same result also when r = 4, for q = 20, and when r = 5, for all q ≧ 14 except multiples of three.     

Bounded transversals in multipartite graphs

Transversals in r‐partite graphs with various properties are known to have many applications in graph theory and theoretical computer science. We investigate f‐bounded transversal s (or f‐BT), that is, transversals whose connected components have order at most f. In some sense we search for the sparsest f‐BT‐free graphs. We obtain estimates on the smallest maximum degree that 3‐partite and 4‐partite graphs without 2‐BT can have and provide a greatly simplified proof of the best known general lower bound on the smallest maximum degree in f‐BT‐free graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory.     

Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole

We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid G _n of order n is similar to the disjoint union of two copies of the quartered Aztec diamond QAD _n−1 of order n−1 with the path P _n ⁽²⁾ on n vertices having edge weights equal to 2. Our proof is based on an explicit change of basis in the vector space on which the adjacency matrix acts. The arguments verifying that this change of basis works are combinatorial. It follows in particular that the characteristic polynomials of the above graphs satisfy the equality P(G _n )=P(P _n ⁽²⁾)[P(QAD _n−1)]². On the one hand, this provides a combinatorial explanation for the “squarishness” of the characteristic polynomial of the square grid—i.e., that it is a perfect square, up to a factor of relatively small degree. On the other hand, as formulas for the characteristic polynomials of the path and the square grid are well known, our equality determines the characteristic polynomial of the quartered Aztec diamond. In turn, the latter allows computing the number of spanning trees of quartered Aztec diamonds. We present and analyze three more families of graphs that share the above described “linear squarishness” property of square grids: odd Aztec diamonds, mixed Aztec diamonds, and Aztec pillowcases—graphs obtained from two copies of an Aztec diamond by identifying the corresponding vertices on their convex hulls. We apply the above results to enumerate all the symmetry classes of spanning trees of the even Aztec diamonds, and all the symmetry classes not involving rotations of the spanning trees of odd and mixed Aztec diamonds. We also enumerate all but the base case of the symmetry classes of perfect matchings of odd square grids with the central vertex removed. In addition, we obtain a product formula for the number of spanning trees of Aztec pillowcases. Research supported in part by NSF grant DMS-0500616.     

Explicit constructions of graphs without short cycles and low density codes

We give an explicit construction of regular graphs of degree 2r withn vertices and girth ≧c logn/logr. We use Cayley graphs of factor groups of free subgroups of the modular group. An application to low density codes is given.     

On the Chromatic Number of H-Free Graphs of Large Minimum Degree

The problem of determining the chromatic number of H-free graphs has been well studied, with particular attention to K _r -free graphs with large minimum degree. Recent progress has been made for triangle-free graphs on n vertices with minimum degree larger than n/3. In this paper, we determine the family of r-colorable graphs H_r{\mathcal{H}_r}, such that if H ? H_r{H \in \mathcal{H}_r}, there exists a constant C < (r − 2)/(r − 1) such that any H-free graph G on n vertices with δ(G) > Cn has chromatic number bounded above by a function dependent only on H and C. A value of C < (r − 2)/(r − 1) is given for every H ? H_r{H \in \mathcal{H}_r}, with particular attention to the case when χ(H) = 3.     

Isomorphic factorizations VII. Regular graphs and tournaments

It is shown using enumeration results that for r > 2t, almost all labeled r-regular graphs cannot be factorized into t ? 2 isomorphic subgraphs. However, no examples of such nonfactorizable graphs are known which satisfy the obvious divisibility condition that the number of edges is divisible by t. Similar observations hold for regular tournaments (t ? 2} and for r-regular digraphs (r > t ? 2).