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. The tripartite-circle crossing number of a triparti
te graph is the minimum number of edge crossings among all tripartite-circle drawings. We determine the tripartite-circle crossing number of $K_{2,2,n}$.
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 nu
mber 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.
We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way that, if at
each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set $R$ of $r$ red points and a set $B$ of $b$ blue points in the plane in general position, how many points of $Rcup B$ can be $K_{1,3}$-covered? and we prove the following results: (1) If $r=3g+h$ and $b=3h+g$, for some non-negative integers $g$ and $h$, then there are point sets $Rcup B$, like ${1,3}$-equitable sets (i.e., $r=3b$ or $b=3r$) and linearly separable sets, that can be $K_{1,3}$-covered. (2) If $r=3g+h$, $b=3h+g$ and the points in $Rcup B$ are in convex position, then at least $r+b-4$ points can be $K_{1,3}$-covered, and this bound is tight. (3) There are arbitrarily large point sets $Rcup B$ in general position, with $r=b+1$, such that at most $r+b-5$ points can be $K_{1,3}$-covered. (4) If $ble rle 3b$, then at least $frac{8}{9}(r+b-8)$ points of $Rcup B$ can be $K_{1,3}$-covered. For $r>3b$, there are too many red points and at least $r-3b$ of them will remain uncovered in any $K_{1,3}$-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.
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 $.
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 leftlfloorfrac{mathstrut n}{mathstrut 2}rightrfloor leftlfloorfrac{mathstrut n-1}{mathstrut 2}rightrfl
oor leftlfloorfrac{mathstrut n-2}{mathstrut 2}rightrfloor leftlfloorfrac{mathstrut n-3}{mathstrut 2}right rfloor$. Abrego et al. introduced the notion of shellability of a drawing $D$ of $K_n$. They proved that if $D$ is $s$-shellable for some $sgeqlfloorfrac{n}{2}rfloor$, then $D$ has at least $H(n)$ crossings. This is the first combinatorial condition on a drawing that guarantees at least $H(n)$ crossings. In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b leq s-2$, also $b$-bishellable. Our main result is that $(lfloor frac{n}{2} rfloor!-!2)$-bishellability of a drawing $D$ of $K_n$ also guarantees, with a simpler proof than for $s$-shellability, that $D$ has at least $H(n)$ crossings. We exhibit a drawing of $K_{11}$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $sgeq5$. This shows that we have properly extended the class of drawings for which the Harary-Hill Conjecture is proved. Moreover, we provide an infinite family of drawings of $K_n$ that are $(lfloor frac{n}{2} rfloor!-!2)$-bishellable, but not $s$-shellable for any $sgeqlfloorfrac{n}{2}rfloor$.
