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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 $, denoted by $ u_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph on $n$ vertices. We improve the lower bounds on $ u_k(K_n) $ for all $ kgeq 14 $ and determine $ u_k(K_n) $ whenever $ 2 < n/k leq 3 $. Our proofs rely on bounding the number of edges in convex graphs with small local crossing numbers. In particular, we determine the maximum number of edges that a convex graph with local crossing number at most $ ell $ can have for $ ellleq 4 $.
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Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,...,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=si_1si_2 cdotssi_d$ of a Coxeter element $c$ of $W$, such that $si_i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,...,d$, and such that the factorisation is minimal in the sense that the sum of the ranks of the $T_i$s, $i=1,2,...,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author. The type $A_n$ decomposition numbers have been computed by Goulden and Jackson, albeit using a somewhat different language. We explain how to extract the type $B_n$ decomposition numbers from results of Bona, Bousquet, Labelle and Leroux on map enumeration. Our formula for the type $D_n$ decomposition numbers is new. These results are then used to determine, for a fixed positive integer $l$ and fixed integers $r_1le r_2le ...le r_l$, the number of multi-chains $pi_1le pi_2le ...le pi_l$ in Armstrongs generalised non-crossing partitions poset, where the poset rank of $pi_i$ equals $r_i$, and where the block structure of $pi_1$ is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type $D_n$ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrongs $F=M$ Conjecture in type $D_n$.
Let $B_n^{(k)}$ be the book graph which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this paper, we first determine the exact value of $r(B_n^{(2)}, C_m)$ for $frac{8}{9}n+112le mle lceilfrac{3n}{2}rceil+1$ and $n geq 1000$. This answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey numbers, {it Ars Combin.,} {bf 31} (1991), 239--248) in a stronger form when $m$ and $n$ are large. Building upon this exact result, we are able to determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each $k geq 3$. Namely, we prove that for each $k geq 3$, $r(B_n^{(k)}, C_n)= (k+1+o_k(1))n.$ This extends a result due to Rousseau and Sheehan (A class of Ramsey problems involving trees, {it J.~London Math.~Soc.,} {bf 18} (1978), 392--396).
We introduce a model for random geodesic drawings of the complete bipartite graph $K_{n,n}$ on the unit sphere $mathbb{S}^2$ in $mathbb{R}^3$, where we select the vertices in each bipartite class of $K_{n,n}$ with respect to two non-degenerate probability measures on $mathbb{S}^2$. It has been proved recently that many such measures give drawings whose crossing number approximates the Zarankiewicz number (the conjectured crossing number of $K_{n,n}$). In this paper we consider the intersection graphs associated with such random drawings. We prove that for any probability measures, the resulting random intersection graphs form a convergent graph sequence in the sense of graph limits. The edge density of the limiting graphon turns out to be independent of the two measures as long as they are antipodally symmetric. However, it is shown that the triangle densities behave differently. We examine a specific random model, blow-ups of antipodal drawings $D$ of $K_{4,4}$, and show that the triangle density in the corresponding crossing graphon depends on the angles between the great circles containing the edges in $D$ and can attain any value in the interval $bigl(frac{83}{12288}, frac{128}{12288}bigr)$.
A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. We present upper and lower bounds on the minimum number of crossings in tripartite-circle drawings of $K_{m,n,p}$ and the exact value for $K_{2,2,n}$. In contrast to 1- and 2-circle drawings, which may attain the Harary-Hill bound, our results imply that balanced restricted 3-circle drawings of the complete graph are not optimal.
Let $mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of $C_4$, $K_{2,t}$, $K_{3,3}$ or $K_{s,t}$ when $t>s!$.
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