Bounding the crossing number of a graph in terms of the crossing number of a minor with small maximum degree

We show that if G has a minor M with maximum degree at most 4, then the crossing number of G in a surface Σ is at least one fourth the crossing number of M in Σ. We use this result to show that every graph embedded on the torus with representativity r ≥ 6 has Klein bottle crossing number at least ⌊2r/3⌋²/64. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 168–173, 2001     

Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method

The biplanar crossing number cr₂(G) of a graph G is min{cr(G₁) + cr(G₂)}, where cr is the planar crossing number. We show that cr₂(G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) ‐ 2 ≤ Kcr₂(G)^0.4057 log₂n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for any size exceeding a certain threshold, there exists a graph G of this size, which simultaneously has the following properties: cr(G) is roughly as large as it can be for any graph of that size, and cr₂(G) is as small as it can be for any graph of that size. The existence is shown using the probabilistic method. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Skewness of generalized Petersen graphs and related graphs

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. In this paper, we determine the skewness of the generalized Petersen graph P(4k, k) and hence a lower bound for the crossing number of P(4k, k). In addition, an upper bound for the crossing number of P(4k, k) is also given.     

Crossing numbers of imbalanced graphs

The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. According to the Crossing Lemma of M. Ajtai, V. Chvátal, M. Newborn, E. Szemerédi, Theory and Practice of Combinatorics, North‐Holland, Amsterdam, New York, 1982, pp. 9–12 and F. T. Leighton, Complexity Issues in VLSI, MIT Press, Cambridge, 1983, the crossing number of any graph with n vertices and e>4n edges is at least constant times e³/n². Apart from the value of the constant, this bound cannot be improved. We establish some stronger lower bounds under the assumption that the distribution of the degrees of the vertices is irregular. In particular, we show that if the degrees of the vertices are d₁?d₂?···?d_n, then the crossing number satisfies \begin{eqnarray*}{\rm{cr}}(G)\geq \frac{c_{1}}{n}\end{eqnarray*} with \begin{eqnarray*}{\textstyle\sum\nolimits_{{{i}}={{{1}}}}^{{{n}}}}{{id}}_{{{i}}}^{{{3}}}-{{c}}_{{{2}}}{{n}}^{{{2}}}\end{eqnarray*}, and that this bound is tight apart from the values of the constants c₁, c₂>0. Some applications are also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 12–21, 2010     

Trading crossings for handles and crosscaps

Let c_k = cr_k (G) denote the minimum number of edge crossings when a graph G is drawn on an orientable surface of genus k. The (orientable) crossing sequence c_o, c₁,c₂…encodes the trade‐off between adding handles and decreasing crossings. We focus on sequences of the type c_o > c₁ > c₂ = 0; equivalently, we study the planar and toroidal crossing number of doubly‐toroidal graphs. For every ? > 0 we construct graphs whose orientable crossing sequence satisfies c₁/c_o > 5/6??. In other words, we construct graphs where the addition of one handle can save roughly 1/6th of the crossings, but the addition of a second handle can save five times more crossings. We similarly define the non‐orientable crossing sequence ?₀,?₁,?₂, ··· for drawings on non‐orientable surfaces. We show that for every ?₀ > ?₁ > 0 there exists a graph with non‐orientable crossing sequence ?₀, ?₁, 0. We conjecture that every strictly‐decreasing sequence of non‐negative integers can be both an orientable crossing sequence and a non‐orientable crossing sequence (with different graphs). © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 230–243, 2001     

Characterizing Graphs with Crossing Number at Least 2

Our main result includes the following, slightly surprising, fact: a 4‐connected nonplanar graph G has crossing number at least 2 if and only if, for every pair of edges having no common incident vertex, there are vertex‐disjoint cycles in G with one containing e and the other containing f.     

Which Crossing Number Is It Anyway?

A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the odd-crossing number) of G is the minimum number of pairs of edges that cross (resp. cross an odd number of times) over all drawings of G. We prove that the largest of these numbers (the crossing number) cannot exceed, twice the square of the smallest (the odd-crossing number). Our proof is based on the following generalization of an old result of Hanani, which is of independent interest. Let G be a graph and let E₀ be a subset of its edges such that there is a drawing of G, in which every edge belonging to E₀ crosses any other edge an even number of times. Then g can be redrawn so that the elements of E₀ are not involved in any crossing. Finally, we show that the determination of each of these parameters is an NP-hard problem and it is NP-complete in the case of the crossing number and the odd-crossing number.     

Triple Crossing Numbers of Graphs

We introduce the triple crossing number,a variation of the crossing number,of a graph,which is the minimal number of crossing points in all drawings of the graph with only triple crossings.It is defined to be zero for planar graphs,and to be infinite for non-planar graphs which do not admit a drawing with only triple crossings.In this paper,we determine the triple crossing numbers for all complete multipartite graphs which include all complete graphs.     

Unexpected behaviour of crossing sequences

The nth crossing number of a graph G, denoted ncr(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which ₀cr(G)=a, ₁cr(G)=b, and ₂cr(G)=0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.     

On Crossing Numbers of Complete Tripartite and Balanced Complete Multipartite Graphs

The crossing number cr(G) of a graph G is the minimum number of crossings in a drawing of G in the plane with no more than two edges intersecting at any point that is not a vertex. The rectilinear crossing number of G is the minimum number of crossings in a such drawing of G with edges as straight line segments. Zarankiewicz proved in 1952 that . We generalize the upper bound to and prove . We also show that for n large enough, and , with the tighter rectilinear lower bound established through the use of flag algebras. A complete multipartite graph is balanced if the partite sets all have the same cardinality. We study asymptotic behavior of the crossing number of the balanced complete r‐partite graph. Richter and Thomassen proved in 1997 that the limit as of over the maximum number of crossings in a drawing of exists and is at most . We define and show that for a fixed r and the balanced complete r‐partite graph, is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.     

An improvement of the crossing number bound

The crossing number cr(G) of a simple graph G with n vertices and m edges is the minimum number of edge crossings over all drawings of G on the ?² plane. The conjecture made by Erd?s in 1973 that cr(G) ≥ Cm³/n² was proved in 1982 by Leighton with C = 1/100 and this constant was gradually improved to reach the best known value C = 1/31.08 obtained recently by Pach, Radoic?i?, Tardos, and Tóth [4] for graphs such that m ≥ 103n/16. We improve this result with values for the constant in the range 1/31.08 ≤ C &< 1/15 where C depends on m/n². For example, C > 1/25 for graphs with m/n² > 0.291 and n > 22, and C > 1/20 for dense graphs with m/n² ≥ 0.485. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On chromatic‐choosable graphs

A graph is chromatic‐choosable if its choice number coincides with its chromatic number. It is shown in this article that, for any graph G, if we join a sufficiently large complete graph to G, then we obtain a chromatic‐choosable graph. As a consequence, if the chromatic number of a graph G is close enough to the number of vertices in G, then G is chromatic‐choosable. We also propose a conjecture related to this fact. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 130–135, 2002     

Note on the Pair-crossing Number and the Odd-crossing Number

The crossing number of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number is the minimum number of pairs of edges that cross an odd number of times. Clearly, . We construct graphs with . This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte. Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k ²/log ² k). G. Tóth’s research was supported by the Hungarian Research Fund grant OTKA-K-60427 and the Research Foundation of the City University of New York.     

Nearly light cycles in embedded graphs and crossing‐critical graphs

We find a lower bound for the proportion of face boundaries of an embedded graph that are nearly light (that is, they have bounded length and at most one vertex of large degree). As an application, we show that every sufficiently large k‐crossing‐critical graph has crossing number at most 2k + 23. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 151–156, 2006     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

On the crossing numbers of Cartesian products with trees

Zip product was recently used in a note establishing the crossing number of the Cartesian product K₁,n □ P_m. In this article, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to weaken the connectivity condition under which the crossing number is additive for the zip product. Next, we deduce a general theorem for bounding the crossing numbers of (capped) Cartesian product of graphs with trees, which yields exact results under certain symmetry conditions. We apply this theorem to obtain exact and approximate results on crossing numbers of Cartesian product of various graphs with trees. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 287–300, 2007     

Crossing numbers of graph embedding pairs on closed surfaces

We shall prove that any two graphs G₁ and G₂ can be embedded together on a closed surface of genus g with at most 4g · β(G₁) · β(G₂) crossing points on their edges if they are embeddable on the surface, where β(G) stands for the Betti number of G, and show several observations on crossings of graph embedding pairs. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 8–23, 2001     

Graphs with at most one crossing

The crossing number of a graph $G$ is the least number of crossings over all possible drawings of $G$. We present a structural characterization of graphs with crossing number one.     

The book crossing number of a graph

Let G be a graph on n vertices and m edges. The book crossing number of G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page book and edges are drawn on pages, such that each edge is contained by one page. Our main results are two polynomial time algorithms to generate near optimal drawing of G on books. The first algorithm give an O(log² n) times optimal solution, on small number of pages, under some restrictions. This algorithm also gives rise to the first polynomial time algorithm for approximating the rectilinear crossing number so that the coordinates of vertices in the plane are small integers, thus resolving a recent open question concerning the rectilinear crossing number. Moreover, using this algorithm we improve the best known upper bounds on the rectilinear crossing number. The second algorithm generates a drawing of G with O(m²/k²) crossings on k pages. This is within a constant multiplicative factor from our general lower bound of Ω(m³/n²k²), provided that m = Ψ(n²). © 1996 John Wiley & Sons, Inc.     

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.