Mathematics

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for integer polytopes in terms of the length of the description of the polytope (in bits) and the minimum angle between facets of its polar. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure 1?�o(1) and polynomial diameter. Both bounds rely on spectral gaps -- of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second -- which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.

A lattice L is said lowly finite if the set [0,a] is finite for every element a of L . We mainly aim to provide a complete proof that, if M is a subset of a complete lowly finite distributive lattice L containing its join-irreducible elements, and a an element of M which is not join-irreducible, then ??b?�M?�[0,a] μ M (b,a)b belongs to the submodule ?�a?�b+a?�b?�a?�b | a,b?�L??of ZL . That property was originally established by Zaslavsky for finite distributive lattice. It is essential to prove the fundamental theorem of dissection theory as will be seen. We finish with a concrete application of that theorem to face counting for submanifold arrangements.

A plane graph is called a rectangular graph if each of its edges can be oriented either horizontally or vertically, each of its interior regions is a four-sided region and all interior regions can be fitted in a rectangular enclosure. Only planar graphs can be dualized. If the dual of a plane graph is a rectangular graph, then the plane graph is a rectangularly dualizable graph. In 1985, Koźmi?ski and Kinnen presented a necessary and sufficient condition for the existence of a rectangularly dualizable graph for a separable connected plane graph. In this paper, we present a counter example for which the conditions given by them for separable connected plane graphs fail and hence, we derive a necessary and sufficient condition for a plane graph to be a rectangularly dualizable graph.

Let G be a connected d -regular graph with a given order and the second largest eigenvalue λ 2 (G) . Mohar and O (private communication) asked a challenging problem: what is the best upper bound for λ 2 (G) which guarantees that κ(G)?�t+1 , where 1?�t?�d?? and κ(G) is the vertex-connectivity of G , which was also mentioned by Cioab?. As a starting point, we solve this problem in the case t=1 , and characterize all families of extremal graphs.

By coloring a signed graph by signed colors, one obtains the signed chromatic polynomial of the signed graph. For each signed graph we construct graded cohomology groups whose graded Euler characteristic yields the signed chromatic polynomial of the signed graph. We show that the cohomology groups satisfy a long exact sequence which corresponds to signed deletion-contraction rule. This work is motivated by Helme-Guizon and Rong's construction of the categorification for the chromatic polynomial of unsigned graphs.

In this article, a combinatorial characterization of the family of planes of $\PG(3,q)$ which meet a hyperbolic quadric in an irreducible conic, using their intersection properties with the points and lines of $\PG(3,q)$, is given.

In these notes we present a closed-formula solution to the problem of decomposing traces of Lie algebra generators into symmetrized traces and structure constants. The solution is written in terms of Solomon idempotents and exploits a projection derived by Solomon in his work on the Poincare-Birkhoff-Witt theorem.

We consider in detail the well-known family of graphs G(q,t) that establish an asymptotic lower bound for Turán numbers ex(n, K 2,t+1 ) . We prove that G(q,t) for some specific q and t also gives an asymptotic bound for K 3,3 and for some higher complete bipartite graphs as well. The asymptotic bounds we prove are the same as provided by the well-known Norm-graphs.

Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. A bipartition of n is an ordered pair of partitions ( ? 1 , ? 2 ) with the sum of all of the parts being n . In this paper, we introduce a generalized crank named the pd -crank for bipartitions with designated summands and give some inequalities for the pd -crank of bipartitions with designated summands modulo 2 and 3. We also define the pd -crank moments weighted by the parity of pd -cranks μ 2k,bd (??,n) and show the positivity of (?? ) n μ 2k,bd (??,n) . Let M bd (m,n) denote the number of bipartitions of n with designated summands with pd -crank m . We prove a monotonicity property of pd -cranks of bipartitions with designated summands and find that the sequence { M bd (m,n) } |m|?�n is unimodal for n??,5,7 .

We prove that a curious generating series identity implies Faber's intersection number conjecture (by showing that it implies a combinatorial identity already given in arXiv:1902.02742) and give a new proof of Faber's conjecture by directly proving this identity.