相关论文: The 2-page crossing number of $K_n$

In the 1950's, English painter Anthony Hill described drawings of complete graphs $K_n$ in the plane having precisely $$H(n) = \tfrac{1}{4}\lfloor \tfrac{n}{2}\rfloor \, \lfloor \tfrac{n-1}{2}\rfloor \, \lfloor \tfrac{n-2}{2}\rfloor…

In 1958, Hill conjectured that the minimum number of crossings in a drawing of $K_n$ is exactly $Z(n) = \frac{1}{4} \lfloor\frac{n}{2}\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor…

Hill's Conjecture states that the crossing number $\text{cr}(K_n)$ of the complete graph $K_n$ in the plane (equivalently, the sphere) is…

Borrowing L\'aszl\'o Sz\'ekely's lively expression, we show that Hill's conjecture is "asymptotically at least 98.5% true". This long-standing conjecture states that the crossing number cr($K_n$) of the complete graph $K_n$ is $H(n) :=…

It has been long conjectured that the crossing numbers of the complete bipartite graph K_{m,n} and of the complete graph K_n equal Z(m,n) (the value conjectured by Zarankiewicz, who came up with a drawing reaching this value) and Z(n)…

A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $ halfplanes with common boundary $ l $, a line, where the vertices are on $ l $ and the edges cannot cross $ l $. The $ k $-page book crossing number of the graph $ G…

The Harary-Hill Conjecture States that the number of crossings in any drawing of the complete graph $ K_n $ in the plane is at least $Z(n):=\frac{1}{4}\left\lfloor \frac{n}{2}\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor…

A "book with k pages" consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine,…

In 1972, Kainen proved a general lower bound on the crossing number of a graph in a closed surface and conjectured that this bound is tight when the graph is either a complete graph or a complete bipartite graph, and the surface is of genus…

The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is $ H(n) = \frac 1 4 \left\lfloor\frac{\mathstrut n}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut…

A "book" with k pages consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine,…

The Harary-Hill Conjecture states that for $n\geq 3$ every drawing of $K_n$ has at least \begin{align*} H(n) :=…

The Harary-Hill conjecture states that for every $n>0$ the complete graph on $n$ vertices $K_n$, the minimum number of crossings over all its possible drawings equals \begin{align*} H(n) :=…

It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states…

The crossing number of a graph $G$ is the minimum number of pairwise intersections of edges among all drawings of $G$. In this paper, we study the crossing number of $K_{n,n}-nK_2$, $K_n\times P_2$, $K_n\times P_3$ and $K_n\times C_4$.

The number $Z(n):=\lfloor n/2\rfloor\lfloor (n-1)/2\rfloor$ is the smallest number of crossings in a simple planar drawing of $K_{2,n}$ in which both vertices on the 2-side have the same clockwise rotation. For two vertices $u,v$ on the…

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing…

Let $\textrm{cr}(G)$ denote the crossing number of a graph $G$. The well-known Zarankiewicz's conjecture (ZC) asserted $\textrm{cr}(K_{m,n})$ in 1954. In 1971, Harborth gave a conjecture (HC) on $\textrm{cr}(K_{x_1,...,x_n})$. HC on…

We study the \emph{geometric $k$-colored crossing number} of complete graphs $\overline{\overline{\text{cr}}}_k(K_n)$, which is the smallest number of monochromatic crossings in any $k$-edge colored straight-line drawing of $K_n$. We…

Let $Q_n$ be the $n$-dimensional hypercube, and let ${\rm cr}(Q_n)$ be the \textit{crossing number} of $Q_n$. Erd\H{o}s and Guy in 1973 conjectured the following equality: ${\rm cr}(Q_n)=\frac{5}{32}4^n-\lfloor\frac{n^2+1}{2}\rfloor…