On the Maximum Number of Edges in Topological Graphs with no Four Pairwise Crossing Edges

A topological graph is called k -quasi-planar if it does not contain k pairwise crossing edges. It is conjectured that for every fixed k, the maximum number of edges in a k-quasi-planar graph on n vertices is O(n). We provide an affirmative answer to the case k=4.     

On Light Graphs in 3-Connected Plane Graphs Without Triangular or Quadrangular Faces

 We prove that each 3-connected plane graph G without triangular or quadrangular faces either contains a k-path P _k , a path on k vertices, such that each of its k vertices has degree ≤5/3k in G or does not contain any k-path. We also prove that each 3-connected pentagonal plane graph G which has a k-cycle, a cycle on k vertices, k∈ {5,8,11,14}, contains a k-cycle such that all its vertices have, in G, bounded degrees. Moreover, for all integers k and m, k≥ 3, k∉ {5,8,11,14} and m≥ 3, we present a graph in which every k-cycle contains a vertex of degree at least m. Received: June 29, 1998 Final version received: April 11, 2000     

On the Thomassen's conjecture*

C. Thomassen proposed a conjecture: Let G be a k‐connected graph with the stability number α ≥ k, then G has a cycle C containing k independent vertices and all their neighbors. In this paper, we will obtain the following result: Let G be a k‐connected graph with stability number α = k + 3 and C any longest cycle of G, then C contains k independent vertices and all their neighbors. This solves Thomassen's conjecture for the case α = k + 3. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 168–180, 2001     

The property of having a k ‐regular subgraph has a sharp threshold

In this paper it is proved that in the random graph model G(n,p), the property of containing a k ‐regular subgraph, has a sharp threshold for k ≥ 3. It is also shown how to use similar methods to prove, quite easily, the (known fact of) sharpness of having a non empty k ‐core for k ≥ 3. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 509–519, 2013     

Bicolored Matchings in Some Classes of Graphs

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings M_k exist with |M_k∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k.     

The set of values of b for which there exist block designs with b blocks

Let B denote the set of values of b for which there exists a block design with b blocks and for k3, let B_k denote the subset of B determined by the designs with block size k. We present some information about B and the sets B_k. In particular, we discuss, for certain integers h, the question as to whether there exist integers k and k^′ such that the equation b^′=b+h has infinitely many solutions b,b^′ satisfying bB_k and b^′B_k^′. The study is restricted to the case λ=1.     

Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least k

Let G_n,m,k denote the space of simple graphs with n vertices, m edges, and minimum degree at least k, each graph G being equiprobable. Let G have property A_k, if G contains ⌊(k − 1)/2⌋ edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size ⌊n/2⌋. We prove that, for k ≥ 3, there is a constant C_k such that if 2m ≥ C_kn then A_k occurs in G_n,m,k with probability tending to 1 as n → ∞. © 2000 John Wiley & Sons, Inc. J. Graph Theory 34: 42–59, 2000     

Quasi‐randomness of graph balanced cut properties

Quasi‐random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi‐randomness of graphs. Let k ≥ 2 be a fixed integer, α₁,…,α_k be positive reals satisfying \begin{align*}\sum_{i} \alpha_i = 1\end{align*} and (α₁,…,α_k)≠(1/k,…,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V ₁,…,V _k of size α₁n,…,α_kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi‐random. However, the method of quasi‐random hypergraphs they used did not provide enough information to resolve the case (1/k,…,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi‐random. Janson also posed the same question in his study of quasi‐randomness under the framework of graph limits. In this paper, we positively answer their question. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Regular graphs constructed from the classical generalized quadrangle Q(4, q)

Let c_k be the smallest number of vertices in a regular graph with valency k and girth 8. It is known that c_{k + 1}?2(1 + k + k² + k³) with equality if and only if there exists a finite generalized quadrangle of order k. No such quadrangle is known when k is not a prime power. In this case, small regular graphs of valency k + 1 and girth 8 can be constructed from known generalized quadrangles of order q>k by removing a part of its structure. We investigate the case when q = k + 1 is a prime power, and try to determine the smallest graph under consideration that can be constructed from a generalized quadrangle of order q. This problem appears to be much more difficult than expected. We have general bounds and improve these for the classical generalized quadrangle Q(4, q), q even. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:70‐83, 2010     

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.     

Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

A graph G is (k,0)‐colorable if its vertices can be partitioned into subsets V₁ and V₂ such that in G[V₁] every vertex has degree at most k, while G[V₂] is edgeless. For every integer k?0, we prove that every graph with the maximum average degree smaller than (3k+4)/(k+2) is (k,0)‐colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)‐colorable. On the other hand, we construct planar graphs with girth 6 that are not (k,0)‐colorable for arbitrarily large k. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:83–93, 2010     

Polychromatic colorings of bounded degree plane graphs

A polychromatic k‐coloring of a plane graph G is an assignment of k colors to the vertices of G such that every face of G has all k colors on its boundary. For a given plane graph G, one seeks the maximum number k such that G admits a polychromatic k ‐coloring. In this paper, it is proven that every connected plane graph of order at least three, and maximum degree three, other than K₄ or a subdivision of K₄ on five vertices, admits a 3‐coloring in the regular sense (i.e., no monochromatic edges) that is also a polychromatic 3‐coloring. Our proof is constructive and implies a polynomial‐time algorithm. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 269‐283, 2009     

Topological Graphs with No Large Grids

Let G be a topological graph with n vertices, i.e., a graph drawn in the plane with edges drawn as simple Jordan curves. It is shown that, for any constants k,l, there exists another constant C(k,l), such that if G has at least C(k,l)n edges, then it contains a k×l-gridlike configuration, that is, it contains k+l edges such that each of the first k edges crosses each of the last l edges. Moreover, one can require the first k edges to be incident to the same vertex. Received: April, 2003 Janos Pach and Micha Sharir has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, and by a joint grant from the U.S.–Israel Binational Science Foundation. János Pach has also been supported by PSC-CUNY Research Award 63382-0032 and by OTKA T-032452. Micha Sharir has also been supported by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Géza Tóth has been supported by OTKA-T-038397 and by an award from the New York University Research Challenge Fund.     

Structures and Chromaticity of Extremal 3-Colourable Sparse Graphs

 Assume that G is a 3-colourable connected graph with e(G) = 2v(G) −k, where k≥ 4. It has been shown that s ₃(G) ≥ 2 ^{k −3}, where s _r (G) = P(G,r)/r! for any positive integer r and P(G, λ) is the chromatic polynomial of G. In this paper, we prove that if G is 2-connected and s ₃(G) < 2 ^{k −2}, then G contains at most v(G) −k triangles; and the upper bound is attained only if G is a graph obtained by replacing each edge in the k-cycle C _k by a 2-tree. By using this result, we settle the problem of determining if W(n, s) is χ-unique, where W(n, s) is the graph obtained from the wheel W _n by deleting all but s consecutive spokes. Received: January 29, 1999 Final version received: April 8, 2000     

On the Existence of Specific Stars in Planar Graphs

Given a graph G, a (k;a,b,c)-star in G is a subgraph isomorphic to a star K_1,3 with a central vertex of degree k and three leaves of degrees a, b and c in G. The main result of the paper is: Every planar graph G of minimum degree at least 3 contains a (k;a,b,c)-star with a≤ b≤ c and (i) k = 3, a≤ 10, or (ii) k = 4, a = 4, 4≤ b≤ 10, or (iii) k = 4, a = 5, 5≤ b≤ 9, or (iv) k = 4, 6≤ a≤ 7, 6≤ b≤ 8, or (v) k = 5, 4≤ a≤ 5, 5≤ b≤ 6 and 5≤ c≤ 7, or (vi) k = 5 and a = b = c = 6.     

Edge intersection graphs of systems of paths on a grid with a bounded number of bends

We answer some of the questions raised by Golumbic, Lipshteyn and Stern and obtain some other results about edge intersection graphs of paths on a grid (EPG graphs). We show that for any d≥4, in order to represent every n vertex graph with maximum degree d as an edge intersection graph of n paths on a grid, a grid of area Θ(n²) is needed. We also show several results related to the classes B_k-EPG, where B_k-EPG denotes the class of graphs that have an EPG representation such that each path has at most k bends. In particular, we prove: For a fixed k and a sufficiently large n, the complete bipartite graph K_m,n does not belong to B_2m−3-EPG (it is known that this graph belongs to B_2m−2-EPG); for any odd integer k we have B_k-EPG B_k+1-EPG; there is no number k such that all graphs belong to B_k-EPG; only 2^O(knlog(kn)) out of all the labeled graphs with n vertices are in B_k-EPG.     

K‐linked graphs with girth condition

Recently, Mader [ 7 ] proved that every 2k‐connected graph with girth g(G) sufficiently large is k‐linked. We show here that g(G ≥ 11 will do unless k = 4,5. If k = 4,5, then g(G) ≥ 19 will do. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 48–50, 2004     

Maximal sets of triangle-factors on v = 6m vertices

We show that the necessary condition m ≤ k ≤ 3m − 1 that there exists a maximal set of k triangle-factors on 6m ≥ 18 vertices is also sufficient, except possibly when k = m. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 235–244, 1998     

A list version of Dirac's theorem on the number of edges in colour‐critical graphs

One of the basic results in graph colouring is Brooks' theorem [R. L. Brooks, Proc Cambridge Phil Soc 37 ( 4 ) 194–197], which asserts that the chromatic number of every connected graph, that is not a complete graph or an odd cycle, does not exceed its maximum degree. As an extension of this result, Dirac [G. A. Dirac, Proc London Math Soc 7(3) ( 7 ) 161–195] proved that every k‐colour‐critical graph (k ≥ 4) on n ≥ k + 2 vertices has at least ½((k ? 1) n + k ? 3) edges. The aim of this paper is to prove a list version of Dirac's result and to extend it to hypergraphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 165–177, 2002; DOI 10.1002/jgt.998