Annals of Combinatorics最新文献

In this work, we consider the class of Cayley graphs known as generalized Paley graphs (GP-graphs for short) given by (Gamma (k,q) = textrm{Cay}({mathbb {F}}_q, {x^k: xin {mathbb {F}}_q^* })), where ({mathbb {F}}_q) is a finite field with q elements, both in the directed and undirected case. Hence (q=p^m) with p prime, (min {mathbb {N}}) and one can assume that (kmid q-1). We first give the connected components of an arbitrary GP-graph. We show that these components are smaller GP-graphs all isomorphic to each other (generalizing Lim and Praeger’s result from 2009 to the directed case). We then characterize those GP-graphs which are disjoint unions of odd cycles. Finally, we show that (Gamma (k,q)) is non-bipartite except for the graphs (Gamma (2^{m-1},2^m)), (m in {mathbb {N}}), which are isomorphic to (K_2 sqcup cdots sqcup K_2), the disjoint union of (2^{m-1}) copies of (K_2).

In 1992, Bitar and Goles introduced the parallel chip-firing game on undirected graphs. Two years later, Prisner extended the game to directed graphs. While the properties of parallel chip-firing games on undirected graphs have been extensively studied, their analogs for parallel chip-firing games on directed graphs have been sporadic. In this paper, we prove the outstanding analogs of the core results of parallel chip-firing games on undirected graphs for those on directed graphs. We find the possible periods of a parallel chip-firing game on a directed simple cycle and introduce the method of Gauss–Jordan elimination on a Laplacian-like matrix to establish a lower bound on the maximum period of a parallel chip-firing game on an orientation of an undirected complete graph and an undirected complete bipartite graph. Finally, we expand the method of motors by Jiang, Scully, and Zhang to directed graphs to show that a binary string s can be the atomic firing sequence of a vertex in a parallel chip-firing game on a strongly connected directed graph if and only if s contains 1 or (s=0).

In this paper, we explore the properties of zeta functions associated with infinite graphs of groups that arise as quotients of cuspidal tree lattices, including all non-uniform arithmetic quotients of the tree of rank-one Lie groups over local fields. Through various examples, we illustrate pairs of non-isomorphic cuspidal tree lattices with the same Ihara zeta function. In addition, we analyze the spectral behavior of a sequence of graphs of groups whose pole-free regions of zeta functions converge towards 0, which also presents an example of arbitrary small exponential error term in counting geodesic formula.

A graph G is k list equitably colorable, if for any given k-uniform list assignment L, G is L-colorable and each color appears on at most (lceil frac{|V(G)|}{k}rceil ) vertices. Kostochka et al. conjectured that if G is a connected graph with maximum degree at least 3, then G is (Delta (G)) list equitably colorable, unless G is a complete graph or is (K_{k,k}) for some odd k. An equitable k-coloring c of G is a mapping c from V(G) to ([k]={1,2,ldots ,k}) such that (c(u)ne c(v)) for each (uvin E(G)), and for each (k_i), (k_j in [k]), (||{v|c(v)=k_i}|-|{w|c(w)=k_j}||le 1). Chen et al. conjectured that each connected graph with maximum degree (Delta ) that is different from the complete graph (K_{Delta +1}), the complete bipartite graph (K_{Delta , Delta }) and an odd cycle admits an equitable coloring with (Delta ) colors. In this paper, we prove that if G is a planar graph without 5-cycles, then G is k list equitably colorable and equitably k-colorable where (kge max {Delta (G),7}).

In this paper, we consider the moments of statistics on conjugacy classes of the colored permutation groups ({mathfrak {S}}_{n,r}=mathbb {Z}_rwr {mathfrak {S}}_n). We first show that any fixed moment of a statistic coincides on all conjugacy classes where all cycles have sufficiently long length. Additionally, for permutation statistics that can be realized via a process we call order-invariant extension, these moments are polynomials in n. Finally, for the descent statistic on the hyperoctahedral group (B_ncong {mathfrak {S}}_{n,2}), we show that its distribution on conjugacy classes without short cycles satisfies a central limit theorem. Our results build on and generalize previous work of Fulman (J Comb Theory Ser A, 1998), Hamaker and Rhoades (arXiv, 2022), and Campion Loth et al. (arXiv, 2023). In particular, our techniques utilize the latter combinatorial framework.

Let ({mathcal {A}}) be a hyperplane arrangement in the d-dimensional vector space ({mathbb {F}}^d). We study one-element extensions ({mathcal {A}}+H_{varvec{alpha },a}) of ({mathcal {A}}) for all ((varvec{alpha },a)in {mathbb {F}}^{d+1}). Their intersection semi-lattices (L({mathcal {A}}+H_{varvec{alpha },a})) and other combinatorial invariants, including Whitney polynomials, characteristic polynomials, Whitney numbers and face numbers, can be classified by the intersection lattice of the induced adjoint arrangement of ({mathcal {A}}). As a byproduct, we further establish order-preserving relations on these combinatorial invariants and obtain a decomposition formula for the characteristic polynomials (chi ({mathcal {A}}+H_{varvec{alpha },a},t)).

The perfect matching complex of a simple graph G is a simplicial complex having facets (maximal faces) as the perfect matchings of G. This article discusses the perfect matching complex of polygonal line tilings and the (left( 2 times nright) )-grid graph in particular. We use tools from discrete Morse theory to show that the perfect matching complex of any polygonal line tiling is either contractible or homotopy equivalent to a wedge of spheres. While proving our results, we also characterize all the matchings of (left( 2 times nright) )-grid graph that cannot be extended to form a perfect matching.

Given a graph G, its genus polynomial is (Gamma _G(x) = sum _{kge 0} g_k(G)x^k), where (g_k(G)) is the number of two-cell embeddings of G in an orientable surface of genus k. The Log-Concavity Genus Distribution (LCGD) Conjecture states that the genus polynomial of every graph is log-concave. It was further conjectured by Stahl that the genus polynomial of every graph has only real roots, however, this was later disproved. We identify several examples of cubic graphs whose genus polynomials, in addition to having at least one non-real root, have a quadratic factor that is non-log-concave when factored over the real numbers.

Eppstein and Frishberg recently proved that the mixing time for the simple random walk on the 1-skeleton of the associahedron is (O(n^3log ^3 n)). We obtain similar rapid mixing results for the simple random walks on the 1-skeleta of the type-B and type-D associahedra. We adapt Eppstein and Frishberg’s technique to obtain the same bound of (O(n^3log ^3 n)) in type B and a bound of (O(n^{13} log ^2 n)) in type D; in the process, we establish an expansion bound that is tight up to logarithmic factors in type B.