An old problem of Erdős: A graph without two cycles of the same length

Constructing extremal triangle-free graphs using integer programming

Contagious sets in dense graphs

We examine several types of visibility graphs in which sightlines can pass through k objects. For k ≥ 1 we bound the maximum thickness of semi-bar k-visibility graphs between ⎡2/3(k + 1) ⎤ and 2k. In addition we show that the maximum number of edges in arc and circle k-visibility graphs on n vertices is at most (k+1)(3n−k−2) for n > 4k+4 and (n 2) for n ≤ 4k+4, while the maximum chromatic number is at most 6k+6. In semi-arc k-visibility graphs on n vertices, we show that the maximum number of edges is (n 2) for n ≤ 3k+3 and at most (k+1)(2n−(k+2)/2) for n > 3k+3, while the maximum chromatic number is at most 4k+4.

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C k denote a cycle of length k, and let C k denote the set of cycles C ℓ, where 3≤ℓ≤k and ℓ and k have the same parity. Erdős and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ C k ) ∼ z(n,F) — here we write f(n) ∼ g(n) for functions f,g: ℕ → ℝ if lim n→∞ f(n)/g(n)=1. They proved this when F ={C 4} by showing that ex(n,{C 4;C 5})∼z(n,C 4). In this paper, we extend this result by showing that if ℓ∈{2,3,5} and k>2ℓ is odd, then ex(n,C 2ℓ ∪{C k }) ∼ z(n,C 2ℓ ). Furthermore, if k>2ℓ+2 is odd, then for infinitely many n we show that the extremal C 2ℓ ∪{C k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2ℓ, and furthermore the asymptotic result does not hold when (ℓ,k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.

On contact graphs of totally separable packings in low dimensions

The Turan number for a graph H, denoted by $$\text {ex}(n,H)$$ , is the maximum number of edges in any simple graph with n vertices which doesn’t contain H as a subgraph. In this paper we find the lower and upper bounds for $$\text { ex}(n,W_{2t+1})$$ . We show that if $$n\ge 4t$$ , then $$\text { ex}(n,W_{2t+1})\ge \left\lfloor \lfloor \frac{2n+t}{4}\rfloor (n+\frac{t-1}{2}-\lfloor \frac{2n+t}{4}\rfloor )\right\rfloor +1.$$ We also show that for sufficiently large n and $$t\ge 5$$ , $$\text { ex}(n,W_{2t+1})\le \frac{ n^2 }{4}+{t-1\over 2}n$$ . Moreover we find the exact value of the Turan number for $$W_9$$ . That is, we show that for sufficiently large n, $$\text { ex}(n,W_9)= \lfloor \frac{n^2}{4}\rfloor +\lceil \frac{3}{4}n\rceil +1$$ .

Let Sn be the set of simple graphs on n vertices in which no two cycles have the same length. A graph G in Sn is called a simple maximum cycle-distributed graph (simple MCD graph) if there exists no graph G′ in Sn with |E(G′)| > |E(G)|. A planar graph G is called a generalized polygon path (GPP) if G* formed by the following method is a path: corresponding to each interior face f of G (G is a plane graph of G) there is a vertex f* of G*; two vertices f* and g* are adjacent in G* if and only if the intersection of the boundaries of the corresponding interior faces of G is a simple path of G. In this paper, we prove that there exists a simple MCD graph on n vertices such that it is a 2-connected graph being not a GPP if and only if n ∈ {10, 11, 14, 15, 16, 21, 22}. We also prove that, by discussing all the natural numbers except for 75 natural numbers, there are exactly 18 natural numbers, for each n of which, there exists a simple MCD graph on n vertices such that it is a 2-connected graph.

Graph minors XXIII. Nash-Williams' immersion conjecture

Let H be a set of graphs. The planar Turán number, exP(n,H), is the maximum number of edges in an n-vertex planar graph which does not contain any member of H as a subgraph. When H={H} has only one element, we usually write exP(n,H) instead. The study of extremal planar graphs was initiated by Dowden (J Graph Theory 83(3):213–230, 2016). He obtained sharp upper bounds for both exP(n,C5) and exP(n,K4). Later on, sharp upper bounds were proved for exP(n,C6) and exP(n,C7). In this paper, we show that exP(n,{K4,C5})≤157(n-2) and exP(n,{K4,C6})≤73(n-2). We also give constructions which show the bounds are sharp for infinitely many n.

This thesis is devoted to crossing patterns of edges in topological graphs. We consider the following four problems: A thrackle is a graph drawn in the plane such that every pair of edges meet exactly once: either at a common endpoint or in a proper crossing. Conway's Thrackle Conjecture says that a thrackle cannot have more edges than vertices. By a computational approach we improve the previously known upper bound of 1.5n on the maximal number of edges in a thrackle with n vertices to 1.428n. Moreover, our method yields an algorithm with a finite running time that for any e > 0 either verifies the upper bound of (1 + e)n on the maximum number of edges in a thrackle or disproves the conjecture. It is not hard to see that any simple graph admits a poly-line drawing in the plane such that each edge is represented by a polygonal curve with at most three bends, and each edge crossings realizes a prescribed angle α. We show that if we restrict the number of bends per edge to two and allow edges to cross in k different angles, a graph on n vertices admitting such a drawing can have at most O(nk2) edges. This generalizes a previous result of Arikushi et al., in which the authors treated a special case of our problem, where k = 1 and the prescribed angle has 90 degrees. The classical result known as Hanani-Tutte Theorem states that a graph is planar if and only if it admits a drawing in the plane in which each pair of non-adjacent edges crosses an even number of times. We prove the following monotone variant of this result, conjectured by J.Pach and G.Toth. If G has an x-monotone drawing in which every pair of independent edges crosses evenly, then G has an x-monotone embedding (i.e. a drawing without crossings) with the same vertex locations. We show several interesting algorithmic consequences of this result. In a drawing of a graph, two edges form an odd pair if they cross each other an odd number of times. A pair of edges is independent (or non-adjacent) if they share no endpoint. For a graph G we let ocr(G) be the smallest number of odd pairs in a drawing of G and let iocr(G) be the smallest number of independent odd pairs in a drawing of G. We construct a graph G with iocr(G) < ocr(G), answering a question by Szekely, and –for the first time– giving evidence that crossings of adjacent edges may not always be trivial to eliminate.

Local algorithms and other wireless network protocols require the underlying network graph to have specific structural properties to guarantee correctness. Two of these properties are connectivity and absence of intersecting links. Assuring only one of these properties is very often possible, either by considering a dense graph, which is very likely connected, but contains many intersections or a sparse graph which contains only few intersections, but is split up into many components. The task is therefore to choose the edges in a given graph in such a way that the intersections are removed while connectivity is preserved. Based on a Poisson point process and the log-normal shadowing model, we analyse the frequency of connected graphs without intersecting links. To further support such graph structure, we also restrict the maximal length of the edges in the network graph. By simulation we observe conditions how the maximal length of the edges in a graph should be chosen to assure the existence of a large component with few intersections.

On the Turán number for the hexagon

A well-known result of Kupitz from 1982 asserts that the maximal number of edges in a convex geometric graph (CGG) on n vertices that does not contain $$k+1$$k+1 pairwise disjoint edges is kn (provided $$n>2k$$n>2k). For $$k=1$$k=1 and $$k=n/2-1$$k=n/2-1, the extremal examples are completely characterized. For all other values of k, the structure of the extremal examples is far from known: their total number is unknown, and only a few classes of examples were presented, that are almost symmetric, consisting roughly of the kn "longest possible" edges of CK(n), the complete CGG of order n. In order to understand further the structure of the extremal examples, we present a class of extremal examples that lie at the other end of the spectrum. Namely, we break the symmetry by requiring that, in addition, the graph admit an independent set that consists of q consecutive vertices on the boundary of the convex hull. We show that such graphs exist as long as $$q \le n-2k$$q≤n-2k and that this value of q is optimal. We generalize our discussion to the following question: what is the maximal possible number f(n, k, q) of edges in a CGG on n vertices that does not contain $$k+1$$k+1 pairwise disjoint edges, and, in addition, admits an independent set that consists of q consecutive vertices on the boundary of the convex hull? We provide a complete answer to this question, determining f(n, k, q) for all relevant values of n, k and q.

On the number of edges in graphs with a given weakly connected domination number

On unique k-factors and unique [1, k]-factors in graphs