Order types are an equivalence relation between point configurations that capture their combinatorial and convexity properties. Let P be a κ-colored sequence of n ≥ d+1 points in general position in ℝ^d. Let ρ be a κ-colored order type on k ≤ d+1 points that has positive density on P; that is, for some constant δ>0, there are δ·nk k-point subsequences of P that have the same order type as ρ and the same color pattern. In this paper we show that there exists a constant c >0 (depending only on d, δ, k and κ) and disjoint subsets X_1,…,X_k of P, each with at least c · n points, such that for every choice of k points x_i ∈ X_i, (x_1,…,x_k) has the same order type and color pattern as ρ.

We show that the 3-symmetric rectilinear and the 3-symmetric pseudolinear crossing numbers of K_33 are equal. Specifically, we prove that sym-cr_3(K_33) = 14 634 = sym-cr_3(K_33).

We show that, up to isomorphism, there is a unique crossing-minimal rectilinear drawing of K18. As a consequence we settle, in the negative, the following question from Aichholzer and Krasser: does there always exist an crossing-minimal drawing of Kn that contains a crossing-minimal drawing of Kn−1?

Let P be a set of n≥1 points in general position in R2. The edge disjointness graph D(P) of P is the graph whose vertices are all the segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. In this note, we give a full characterization of all those edge disjointness graphs that are hamiltonian. More precisely, we shall show that there are exactly 9 order types of P for which D(P) is not hamiltonian. Additionally, from one of these 9 order types, we derive a counterexample to a criterion for the existence of hamiltonian cycles due to A. D. Plotnikov in 1998.

Let $G=(V(G), E(G))$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. Let $S$ be a subset of $V(G)$, and let $B(S)$ be the set of neighbours of $S$ in $V(G) \setminus S$. The differential $\partial(S)$ of $S$ is the number $|B(S)|-|S|$. The maximum value of $\partial(S)$ taken over all subsets $S\subseteq V(G)$ is the differential $\partial(G)$ of $G$. The graph $R{G}$ is defined as the graph obtained from $G$ by adding a new vertex $v_e$ for each $e\in E(G)$, and by joining $v_e$ to the end vertices of $e$. In this paper we study the relationship between $\partial(G)$ and $\partial(R(G))$, and give tight asymptotic bounds for $\partial(R(G))$. We also exhibit some relationships between certain vertex sets of $G$ and $R(G)$ which involve well known graph theoretical parameters.

Let P be a finite set of points in general position in the plane. The disjointness graph of segments D(P) of P is the graph whose vertices are all the closed straight line segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. As usual, we use χ (D(P)) to denote the chromatic number of D(P), and use d(n) to denote the maximum χ (D(P)) taken over all sets P of n points in general position in the plane. In this paper, we show that d(n)=n-2 if and only if n∈{3,4,… ,16} .

Let n be a positive integer multiple of 3. A rectilinear drawing of the complete graph Kn in the plane is 3–symmetric if its underlying point set P is 3–symmetric, that is, if P is the disjoint union of three equal sized sets Q,ρ(Q) and ρ2(Q) such that ρ is a 2π/3 clockwise rotation around a suitable point in the plane. The 3–symmetric rectilinear crossing number sym−cr3‾(Kn) of Kn is the minimum number of crossings in any 3–symmetric rectilinear drawing of Kn. In this paper, we extend these notions to the more general setting of pseudolinear drawings of Kn by defining the corresponding 3–symmetric pseudolinear crossing number sym−cr3˜(Kn) of Kn, and show that sym−cr3˜(K36)=sym−cr3‾(K36)=21174.

Let L be a set of n non-concurrent blue lines and let R be a set of m red lines in the real projective plane. In this note, using elementary geometric arguments, we show that if L∩R=∅ and there is a line from R through every intersection point of lines in L, then m≥411(n−1311). This lower bound improves the previous ones whenever n≥4. Also we give an application of this result.

Let $P$ be a set of $n\geq 3$ points in general position in the plane. The edge disjointness graph $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in $D(P)$ if and only if they are disjoint. We show that the connectivity of $D(P)$ is at least $\binom{\lfloor\frac{n-2}{2}\rfloor}{2}+\binom{\lceil\frac{n-2}{2}\rceil}{2}$, and that this bound is tight for each $n\geq 3$.

Let G=(V,E) be a simple graph with vertex set V and edge set E. Let D be a subset of V, and let B(D) be the set of neighbors of D in V∖D. The differential ∂(D) of D is defined as |B(D)|−|D|. The maximum value of ∂(D) taken over all subsets D⊆V is the differential of G, denoted by ∂(G). The line graph L(G) of G=(V,E) is the graph whose vertex set is E, and two vertices in L(G) are adjacent if and only if their corresponding edges in G have a common end vertex. In this work we prove that ∂(L(G))≥∂(G)−1 for any graph G, and that ∂(L(G))≥∂(G) for any graph G different from a tree. Moreover, we give a characterization of all trees T such that ∂(L(T))=∂(T)−1.

We study c -crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For every fixed pair of integers with c ≥ 13 and d ≥ 1, we give first explicit constructions of c -crossing-critical graphs containing arbitrarily many vertices of degree greater than d . We also show that such unbounded degree constructions do not exist for c ≤ 12, precisely, that there exists a constant D such that every c -crossing-critical graph with c ≤ 12 has maximum degree at most D . Hence, the bounded maximum degree conjecture of c -crossing-critical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values c ≤ 12. 1

Let {p1,…,pn} and {q1,…,qn} be two sets of n labeled points in general position in the plane. We say that these two point sets have the same order type if for every triple of indices (i,j,k), pk is above the directed line from pi to pj if and only if qk is above the directed line from qi to qj. In this paper we give the first non-trivial lower bounds on the number of different order types of n points that can be realized in integer grids of polynomial size.

Let P be a set of n≥3 points in general position in the plane. The edge disjointness graph D(P) of P is the graph whose vertices are the n2 closed straight line segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. In this paper we show that the connectivity of D(P) is at most 7n218+Θ(n), and that this upper bound is asymptotically tight. The proof is based on the analysis of the connectivity of D(Qn), where Qn denotes an n-point set that is almost 3-symmetric.

Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different colors. In this paper we show that if $P$ is the point configuration known as the double chain, with $k$ points in the upper convex chain and $l \ge k$ points in the lower convex chain, then $k+l- \left\lfloor \sqrt{2l+\frac{1}{4}} - \frac{1}{2}\right\rfloor$ colors are needed and that this number is sufficient.

Let $G$ be a graph of order $n$ and let $k\in\{1,\ldots,n-1\}$. The $k$-token graph $F_k(G)$ of $G$, is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. We study the independence and matching numbers of $F_k(G)$. We present a tight lower bound for the matching number of $F_k(G)$ for the case in which $G$ has either a perfect matching or an almost perfect matching. Also, we estimate the independence number for bipartite $k$-token graphs, and determine the exact value for some graphs.

If G = ( V ( G ) , E ( G ) ) is a simple connected graph with the vertex set V ( G ) and the edge set E ( G ) , S is a subset of V ( G ) , and let B ( S ) be the set of neighbors of S in V ( G ) ∖ S . Then, the differential of S ∂ ( S ) is defined as | B ( S ) | − | S | . The differential of G, denoted by ∂ ( G ) , is the maximum value of ∂ ( S ) for all subsets S ⊆ V ( G ) . The graph operator Q ( G ) is defined as the graph that results by subdividing every edge of G once and joining pairs of these new vertices iff their corresponding edges are incident in G. In this paper, we study the relations between ∂ ( G ) and ∂ ( Q ( G ) ) . Besides, we exhibit some results relating the differential ∂ ( G ) and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial \(B(G;x):={\sum}_{k=-n}^{\partial(G)}B_k(G)x^{n+k}\), where Bk(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G; x) and its coefficients. In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.

Neil Sloane showed that the problem of determine the maximum size of a binary code of constant weight 2 that can correct a single adjacent transposition is equivalent to finding the packing number of a certain graph. In this paper we solve this open problem by finding the packing number of the double vertex graph (2-token graph) of a path graph. This double vertex graph is isomorphic to the Sloane's graph. Our solution implies a conjecture of Rob Pratt about the ordinary generating function of sequence A085680.

A plane drawing of a graph is cylindrical if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The cylindrical crossing number of a graph \(G\) is the minimum number of crossings in a cylindrical drawing of \(G\). In his influential survey on the variants of the definition of the crossing number of a graph, Schaefer lists the complexity of computing the cylindrical crossing number of a graph as an open question. In this paper, we prove that the problem of deciding whether a given graph admits a cylindrical embedding is NP-complete, and as a consequence we show that the \(t\)-cylindrical crossing number problem is also NP-complete. Moreover, we show an analogous result for the natural generalization of the cylindrical crossing number, namely the \(t\)-crossing number.

Dirac and Shuster in 1954 exhibited a simple proof of Kuratowski theorem by showing that any 1-crossing-critical edge of G belongs to a Kuratowski subdivision of G. In 1983, Siran extended this result to any 2-crossing-critical edge e with endvertices b and c of a graph G with crossing number at least two, whenever no two blocks of G - b - c contain all its vertices. Calling an edge f of G k-exceptional whenever f is k-crossing-critical and it does not belong to any Kuratowski subgraph of G, he showed that simple 3-connected graphs with k-exceptional edges exist for any k >= 6, and they exist even for arbitrarily large difference of cr(G) - cr(G - f). In 1991, Kochol constructed such examples for any k >= 4, and commented that Sirdn's result holds for any simple graph. Examining the case when two blocks contain all the vertices of G - b - c, we show that graphs with k-exceptional edges exist for any k >= 2, albeit not necessarily simple. We confirm that no such simple graphs with 2-exceptional edges exist by applying the techniques of the recent characterization of 2-crossing-critical graphs to explicitly describe the set of all graphs with 2-exceptional edges and noting they all contain parallel edges. In this context, the paper can be read as an accessible prelude to the characterization of 2-crossing-critical graphs.