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Let $ G $ be a connected graph. If $bar{sigma}(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the proximity, $pi(G)$, of $G$ is defined as the smallest value of $bar{sigma}(v)$ over all vertices $v$ of $G$. We give upper bounds for the proximity of simple triangulations and quadrangulations of given order and connectivity. We also construct simple triangulations and quadrangulations of given order and connectivity that match the upper bounds asymptotically and are likely optimal.
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We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer $b$ there is such an inductive construction of triangulations with $b$ braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with $b$ braces that is linear in $b$. In the case that $b=1$ or $2$ we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in $mathbb{R}^4$ and a class of mixed norms on $mathbb{R}^3$.
Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Omega(n^{1/4})}$ Hamiltonian cycles.
The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plucker relations from geometry. We give a solution to the Plucker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generating series for a general class of transitive ordered factorizations in the symmetric group satisfies the KP hierarchy. These factorizations appear in geometry as encodings of branched covers, and thus by specializing our transitive factorization result, we are able to prove that the generating series for two classes of branched covers satisfies the KP hierarchy. For the first of these, the double Hurwitz series, this result has been previously given by Okounkov. The second of these, that we call the m-hypermap series, contains the double Hurwitz series polynomially, as the leading coefficient in m. The m-hypermap series also specializes further, first to the series for hypermaps and then to the series for maps, both in an orientable surface. For the latter series, we apply one of the KP equations to obtain a new and remarkably simple recurrence for triangulations in a surface of given genus, with a given number of faces. This recurrence leads to explicit asymptotics for the number of triangulations with given genus and number of faces, in recent work by Bender, Gao and Richmond.
An $S$-hypersimplex for $S subseteq {0,1, dots,d}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.
Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cycles or have minimum degree 5. We show that if $G$ has $O(n/{log}_2 n)$ separating 4-cycles then $G$ has $Omega(n^2)$ Hamiltonian cycles, and if $delta(G)ge 5$ then $G$ has $2^{Omega(n^{1/4})}$ Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a double wheel structure, providing further evidence to the above conjecture.
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