Book crossing numbers of the complete graph and small local convex crossing numbers

Abstract

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 $ ν_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 $ ν_k(K_n) $ for all $ k\geq 14 $ and determine $ ν_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 $ \ell\leq 4 $.

Figures (25)

• Figure 1: The graphs $G_{\ell,n}$, $0\leq \ell \leq 4$, which maximize the number of edges among those convex graphs with $n$ vertices and local crossing number $\ell$: (a) $\ell=0$, (b) $\ell=1$, (c) $\ell=2$, (d) $\ell=3$, and (e) $\ell=4$.
• Figure 2: (a-b) The graphs $S_7$ and $S_7'$, the unique subgraphs of $D_7$ with $11$ edges and local crossing number $4$. (c) The graph $S_8$, a subgraph of $D_8$ with $13$ edges and local crossing number $4$.
• Figure 3: Organization of the proof of Theorem \ref{['theorem:epsilons']}.
• Figure 4: (a) Optimal graph for $e^*(n).$ (b) The corners (dark shade) and the polygon (light shade) whose vertices are the crossings in the cycle $C$.
• Figure 5: Possible cycles $C$ with $j=4$.

Theorems & Definitions (30)

• Conjecture 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• proof
• Theorem 6: Equivalent to Theorem \ref{['theorem:RangeGeneralIntro']}
• proof
• Proposition 7
• proof