On 3-choosable planar graphs of girth at least 4

Let G be a plane graph of girth at least 4. Two cycles of G are intersecting if they have at least one vertex in common. In this paper, we show that if a plane graph G has neither intersecting 4-cycles nor a 5-cycle intersecting with any 4-cycle, then G is 3-choosable, which extends one of Thomassen’s results [C. Thomassen, 3-list-coloring planar graphs of girth 5, J. Combin. Theory Ser. B 64 (1995) 101-107].     

Do 3n − 5 edges force a subdivision of K5?

A conjecture of Dirac states that every simple graph with n vertices and 3n ? 5 edges must contain a subdivision of K₅. We prove that a topologically minimal counterexample is 5-connected, and that no minor-minimal counterexample contains K₄ – e. Consequently, Dirac's conjecture holds for all graphs that can be embedded in a surface with Euler characteristic at least ? 2.     

Almost all graphs with 1.44n edges are 3-colorable

We establish a uniform asymptotic approximation of certain probabilities arising in the coupon collector's problem. Then we use this approximation to prove that almost all graphs with n vertices and 1.44 n edges contain no subgraph with minimum degree at least three, and hence are 3-colorable.     

Nonseparating K4‐subdivisions in graphs of minimum degree at least 4

We first prove that for every vertex x of a 4‐connected graph G, there exists a subgraph H in G isomorphic to a subdivision of the complete graph K₄ on four vertices such that is connected and contains x. This implies an affirmative answer to a question of Kühnel whether every 4‐connected graph G contains a subdivision H of K₄ as a subgraph such that is connected. The motor for our induction is a result of Fontet and Martinov stating that every 4‐connected graph can be reduced to a smaller one by contracting a single edge, unless the graph is the square of a cycle or the line graph of a cubic graph. It turns out that this is the only ingredient of the proof where 4‐connectedness is used. We then generalize our result to connected graphs of minimum degree at least 4 by developing the respective motor, a structure theorem for the class of simple connected graphs of minimum degree at least 4. A simple connected graph G of minimum degree 4 cannot be reduced to a smaller such graph by deleting a single edge or contracting a single edge and simplifying if and only if it is the square of a cycle or the edge disjoint union of copies of certain bricks as follows: Each brick is isomorphic to K₃, K₅, K_{2, 2, 2}, , , or one the four graphs , , , obtained from K₅ and K_{2, 2, 2} by deleting the edges of a triangle, or replacing a vertex x by two new vertices and adding four edges to the endpoints of two disjoint edges of its former neighborhood, respectively. Bricks isomorphic to K₅ or K_{2, 2, 2} share exactly one vertex with the other bricks of the decomposition, vertices of degree 4 in any other brick are not contained in any further brick of the decomposition, and the vertices of a brick isomorphic to K₃ must have degree 4 in G and have pairwise no common neighbors outside that brick.     

On circle graphs with girth at least five

Circle graphs with girth at least five are known to be 2-degenerate [A.A. Ageev, Every circle graph with girth at least 5 is 3-colourable, Discrete Math. 195 (1999) 229-233]. In this paper, we prove that circle graphs with girth at least g≥5 and minimum degree at least two contain a chain of g−4 vertices of degree two, which implies Ageev’s result in the case g=5. We then use this structural property to give an upper bound on the circular chromatic number of circle graphs with girth at least g≥5 as well as a precise estimate of their maximum average degree.     

On circle graphs with girth at least five

Circle graphs with girth at least five are known to be 2-degenerate (Ageev, 1999). In this paper, we prove that circle graphs with girth at least g ⩾ 5 contain a vertex of degree at most one, or a chain of g− 4 vertices of degree two, which implies Ageev's result in the case g = 5. We then use this structural property to give an upper bound on the circular chromatic number of circle graphs with girth at least g ⩾ 5 as well as a precise estimate of their maximum average degree.     

On 3-connected minors of 3-connected matroids and graphs

Whittle proved, for k=1,2, that if N is a 3-connected minor of a 3-connected matroid M, satisfying r(M)−r(N)≥k, then there is a k-independent set I of M such that, for every x∈I, si(M/x) is a 3-connected matroid with an N-minor. In this paper, we establish this result for k=3. It is already known that it cannot be extended to greater values of k. But, here we also show that, in the graphic case, with the extra assumption that r(M)−r(N)≥6, we can guarantee the existence of a 4-independent set of M with such a property. Moreover, in the binary case, we show that if r(M)−r(N)≥5, then M has such a 4-independent set or M has a triangle T meeting 3 triads and such that M/T is a 3-connected matroid with an N-minor.     

On 3-regular graphs having crossing number at least 2

We give a planar proof of the fact that if G is a 3-regular graph minimal with respect to having crossing number at least 2, then the crossing number of G is 2.     

On the density of 2‐colorable 3‐graphs in which any four points span at most two edges

Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)$. We improve the previously best known lower and upper bounds of 0.25682 and 3/10?ε, respectively, by showing that This implies the following new upper bound for the Turán density of K In order to establish these results we use a combination of the properties of computer‐generated extremal 3‐graphs for small n and an argument based on “super‐saturation”. Our computer results determine the exact values of ex(n, K) for n≤19 and ex₂(n, K) for n≤17, as well as the sets of extremal 3‐graphs for those n. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 105–114, 2010     

Reductions of 3-connected graphs with minimum degree at least four

We show that if G is a 3-connected graph of minimum degree at least 4 and with |V (G)| ≥ 7 then one of the following is true: (1) G has an edge e such that G/e is a 3-connected graph of minimum degree at least 4; (2) G has two edges uv and xy with ux, vy, vx ∈ E(G) such that the graph G/uv/xy obtained by contraction of edges uv and xy in G is a 3-connected graph of minimum degree at least 4; (3) G has a vertex x with N(x) = {x₁, x₂, x₃, x₄} and x₁x₂, x₃x₄ ∈ E(G) such that the graph (G ? x)/x₁x₂/x₃x₄ obtained by contraction of edges x₁x₂ and x₃x₄ in G – x is a 3-connected graph of minimum degree at least 4.

Each of the three reductions is necessary: there exists an infinite family of 3- connected graphs of minimum degree not less than 4 such that only one of the three reductions may be performed for the members of the family and not the two other reductions.     

Finding for a Surface Σ of Characteristic −4

For each surface Σ, we define max G is a class two graph of maximum degree that can be embedded in . Hence, Vizing's Planar Graph Conjecture can be restated as if Σ is a sphere. In this article, by applying some newly obtained adjacency lemmas, we show that if Σ is a surface of characteristic . Until now, all known satisfy . This is the first case where .     

A Class of 2‐(3n7, 3n−17, (3n−17−1)/2) Designs

Generalized Hadamard matrices are used for the construction of a class of quasi‐residual nonresolvable BIBD's with parameters . The designs are not embeddable as residual designs into symmetric designs if n is even. The construction yields many nonisomorphic designs for every given n ≥ 2, including more than 10¹⁷ nonisomorphic 2‐(63,21,10) designs. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 460–464, 2007