Mathematics

We introduce a polynomial-time algorithm for spanning tree modulus based on Cunningham's algorithm for graph vulnerability. The algorithm exploits an interesting connection between spanning tree modulus and critical edge sets from the vulnerability problem. This paper describes the new algorithm, describes a practical means for implementing it using integer arithmetic, and presents some examples and computational time scaling tests.

This paper proves Lee-Lee's conjecture that establishes a coincidence between the set of associated roots of non-self-intersecting curves in a n -punctured disc and the set of real Schur roots of acyclic (valued) quivers with n vertices.

The Erd?s-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n . In this paper, we prove this conjecture for every large n . We also provide stability versions of this result, which confirm a prediction of Kahn.

We prove the Extended Delta Conjecture of Haglund, Remmel, and Wilson, a combinatorial formula for ? h l ? ??e k e n , where ? ??e k and ? h l are Macdonald eigenoperators and e n is an elementary symmetric function. We actually prove a stronger identity of infinite series of G L m characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.

The permanent-on-top conjecture states that the largest eigenvalue of the Schur power matrix of a positive semi-definite Hermitian matrix H is per(H). A counterexample has been found with the help of computers, but here, I present another counterexample that can be checked by hand. My method is to use linear representations of groups to connect the spectrum of the Schur power matrix with the spectra of the entrywise(Hadamard) product matrices of permanental compound matrices. By that, we are able to study the properties of the spectrum of the Schur power matrix through the entrywise(Hadamard) product matrices of permanental compound matrices. The counterexample we find is in fact also a counterexample to a weaker conjecture related to permanental compound matrices. This conjecture was also known to be false, but the new counterexample is smaller than the known one.

The Color Critical Edge Theorem of Simonovits states that for a given color critical graph H with chromatic number ?(H)=k+1 , there exists an n 0 (H) such that the Turán graph T n,k is the only extremal graph with respect to ex(n,H) provided n??n 0 (H) . Nikiforov's pioneer work on spectral graph theory implies that the color critical edge theorem also holds if ex(n,H) is replaced by the maximum spectral radius and n 0 (H) is an exponential function on |H| . We conjecture that n 0 (H) is a linear function of |H| . Previous evidences include complete graphs and odd cycles. In this paper, we confirm two new classes of graphs: books and theta graphs. Namely, we prove that every graph on n vertices and ?(G)>?( T n,2 ) contains a book of size greater than n 6.5 . This can be seen as a spectral version of a 1962 conjecture by Erd?s, which states that every graph on n vertices and e(G)>e( T n,2 ) contains a book of size greater than n 6 . In addition, our result on theta graphs implies that if G is a graph of order n and ?(G)>?( T n,2 ) , then G contains a cycle of length t for every t??n 7 . This is related to an open question by Nikiforov which asks to determine the maximum c such that if G is a graph of order n and ?(G)>?( T n,2 ) , then G contains a cycle of length t for every t?�cn .

We address a long-standing and long-investigated problem in combinatorial topology, and break the exponential barrier for triangulations of real projective space, constructing a trianglation of RP n of size e ( 1 2 +o(1)) n √ logn .

We prove that for all positive integers n and k , there exists an integer N=N(n,k) satisfying the following. If U is a set of k direction vectors in the plane and J U is the set of all line segments in direction u for some u?�U , then for every N families F 1 ,?? F N , each consisting of n mutually disjoint segments in J U , there is a set { A 1 ,?? A n } of n disjoint segments in ??1?�i?�N F i and distinct integers p 1 ,?? p n ?�{1,??N} satisfying that A j ??F p j for all j?�{1,??n} . We generalize this property for underlying lines on fixed k directions to k families of simple curves with certain conditions.

Erd?s and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example cycles of length at least ??, cycles of length p modulo q , cycles intersecting a prescribed set of vertices at least t times, and cycles contained in given Z 2 -homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.

We present a unifying framework in which both the ν -Tamari lattice, introduced by Préville-Ratelle and Viennot, and principal order ideals in Young's lattice indexed by lattice paths ν , are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the ν -caracol flow polytopes. The first triangulation gives a new geometric realization of the ν -Tamari complex introduced by Ceballos, Padrol and Sarmiento. We use the second triangulation to show that the h ??-vector of the ν -caracol flow polytope is given by the ν -Narayana numbers, extending a result of Mészáros when ν is a staircase lattice path. Our work generalizes and unifies results on the dual structure of two subdivisions of a polytope studied by Pitman and Stanley.