Recently, a conjecture due to Hendry was disproved which stated that every Hamiltonian chordal graph is cycle extendible. Here we further explore the conjecture, showing that it fails to hold even when a number of extra conditions are imposed. In particular, we show that Hendry's Conjecture fails for strongly chordal graphs, graphs with high connectivity, and if we relax the definition of "cycle extendible" considerably. We also consider the original conjecture from a subtree intersection model point of view, showing that a result of Abuieda et al is nearly best possible.
This note describes a SAT encoding for the $n$-fractions puzzle which is problem 041 of the CSPLib. Using a SAT solver we obtain a solution for two of the six remaining open instances of this problem.
T-H. Hubert Chan, Anand Louis, Zhihao Gavin Tang
et al.
The celebrated Cheeger's Inequality establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. In this paper we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is non-linear, we have to exploit other properties of the diffusion process to recover a spectral property concerning the "second eigenvalue" of the resulting Laplacian. Moreover, we show that higher order spectral properties cannot hold in general using the current framework. We consider a stochastic diffusion process, in which each vertex also experiences Brownian noise from outside the system. We show a relationship between the second eigenvalue and the convergence behavior of the process. We show that various hypergraph parameters like multi-way expansion and diameter can be bounded using this operator's spectral properties. Since higher order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher order Cheeger-like inequalities. For any positive integer $k$, we give a polynomial time algorithm to compute an $O(\log r)$-approximation to the $k$-th procedural minimizer, where $r$ is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis for constant values of $k$.
We introduce and study, for a process P delivering edges on the Cartesian product of the vertex sets of a given set of graphs, the P-product of these graphs, thereby generalizing many types of product graph. Analogous to the notion of a multilinear map (from linear algebra), a P-morphism is introduced and utilised to define a P-tensor product of graphs, after which its uniqueness is demonstrated. Congruences of graphs are utilised to show a way to handle projections (being weak homomorphisms) in this context. Finally, the graph of a homomorphism and a P-tensor product of homomorphisms are introduced, studied, and linked to the P-tensor product of graphs.
We give an overview of new and existing cut- and flow-based ILP formulations for the two-stage stochastic Steiner tree problem and compare the strength of the LP relaxations.
We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area; skip; dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q; t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area; skip; dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q; t$-symmetry of $(m,n)$-rational $q; t$-Catalan polynomials for $m=3$..
This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan.
Let G be a connected graph and S be a set of vertices. The h-extra connectivity of G is the cardinality of a minimum set S such that G-S is disconnected and each component of G-S has at least h+1 vertices. The h-extra connectivity is an important parameter to measure the reliability and fault tolerance ability of large interconnection networks. The h-extra connectivity for h=1,2 of k-ary n-cube are gotten by Hsieh et al. in [Theoretical Computer Science, 443 (2012) 63-69] for k>=4 and Zhu et al. in [Theory of Computing Systems, arxiv.org/pdf/1105.0991v1 [cs.DM] 5 May 2011] for k=3. In this paper, we show that the h-extra connectivity of the 3-ary n-cube networks for h=3 is equal to 8n-12, where n>=3.
A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.
We show that every bridgeless cubic graph $G$ on $n$ vertices other than the Petersen graph has a 2-factor with at most $2(n-2)/15$ circuits of length $5$. An infinite family of graphs attains this bound. We also show that $G$ has a 2-factor with at most $n/5.8\overline{3}$ odd circuits. This improves the previously known bound of $n/5.41$ [Luko\v{t}ka, M\'a\v{c}ajov\'a, Maz\'ak, \v{S}koviera: Small snarks with large oddness, arXiv:1212.3641 [cs.DM] ].
Let $(W, S)$ be a Coxeter system. A $W$-graph is an encoding of a representation of the corresponding Iwahori-Hecke algebra. Especially important examples include the $W$-graph corresponding to the action of the Iwahori-Hecke algebra on the Kazhdan-Lusztig basis as well as this graph's strongly connected components (cells). In 2008, Stembridge identified some common features of the Kazhdan-Lusztig graphs ("admissibility'') and gave combinatorial rules for detecting admissible $W$-graphs. He conjectured, and checked up to $n=9$, that all admissible $A_n$-cells are Kazhdan-Lusztig cells. The current paper provides a possible first step toward a proof of the conjecture. More concretely, we prove that the connected subgraphs of $A_n$-cells consisting of simple (i.e. directed both ways) edges do fit into the Kazhdan-Lusztig cells.
In a recent work, the combinatorial interpretation of the polynomial $\alpha (n; k_1,k_2,\ldots,k_n)$ counting the number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ was extended to weakly decreasing sequences $k_1 ≥k_2 ≥⋯≥k_n$. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles – a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of $\alpha (n; k_1,k_2,\ldots,k_n)$ at arbitrary $(k_1,k_2,\ldots,k_n) ∈ \mathbb{Z}^n$ is a signed enumeration of Generalized Monotone Triangles with bottom row $(k_1,k_2,\ldots,k_n)$. Computational experiments indicate that certain evaluations of the polynomial at integral sequences yield well-known round numbers related to Alternating Sign Matrices. The main result provides a combinatorial interpretation of the conjectured identities and could turn out useful in giving bijective proofs.
We study variants of Gale-Robinson sequences, as motivated by cluster algebras with principal coefficients. For such cases, we give combinatorial interpretations of cluster variables using brane tilings, as from the physics literature.
We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.
Let G be a connected graph and S be a set of vertices. The h-extra connectivity of G is the cardinality of a minimum set S such that G-S is disconnected and each component of G-S has at least h+1 vertices. The h-extra connectivity is an important parameter to measure the reliability and fault tolerance ability of large interconnection networks. The h-extra connectivity for h=1,2 of k-ary n-cube are gotten by Hsieh et al. in [Theoretical Computer Science, 443 (2012) 63-69] for k>=4 and Zhu et al. in [Theory of Computing Systems, arxiv.org/pdf/1105.0991v1 [cs.DM] 5 May 2011] for k=3. In this paper, we show that the h-extra connectivity of the 3-ary n-cube networks for h=3 is equal to 8n-12, where n>=3.