Related papers: Parking functions and chip-firing on hypergraphs
Kontsevich conjectured that the number f(G,q) of zeros over the finite field with q elements of a certain polynomial connected with the spanning trees of a graph G is polynomial function of q. We have been unable to settle Kontsevich's…
The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…
The notion of vertex sparsification is introduced in \cite{M}, where it was shown that for any graph $G = (V, E)$ and a subset of $k$ terminals $K \subset V$, there is a polynomial time algorithm to construct a graph $H = (K, E_H)$ on just…
If distinct colours represent distinct technology types that are placed at the vertices of a simple graph in accordance to a minimum proper colouring, a disaster recovery strategy could rely on an answer to the question: "What is the…
We say that a hereditary graph class $\mathcal{G}$ is \emph{clique-sparse} if there is a constant $k=k(\mathcal{G})$ such that for every graph $G\in\mathcal{G}$, every vertex of $G$ belongs to at most $k$ maximal cliques, and any maximal…
The problem of computing all maximal induced subgraphs of a graph G that have a graph property P, also called the maximal P-subgraphs problem, is considered. This problem is studied for hereditary, connected-hereditary and rooted-hereditary…
We consider straight line drawings of a planar graph $G$ with possible edge crossings. The \emph{untangling problem} is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let $fix(G)$ denote the maximum…
An $h$-queue layout of a graph $G$ consists of a linear order of its vertices and a partition of its edges into $h$ queues, such that no two independent edges of the same queue nest. The minimum $h$ such that $G$ admits an $h$-queue layout…
If $G$ is a graph and $\mathcal{H}$ is a set of subgraphs of $G$, we say that an edge-coloring of $G$ is $\mathcal{H}$-polychromatic if every graph from $\mathcal{H}$ gets all colors present in $G$ on its edges. The…
Let $G$ be a graph and let $l$ be an integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the…
Parking sequences (a generalization of parking functions) are defined by specifying car lengths and requiring that a car attempts to park in the first available spot after its preference. If it does not fit there, then a collision occurs…
Given a positive-integer-valued vector $u=(u_1, \dots, u_m)$ with $u_1<\cdots<u_m$. A $u$-parking function of length $m$ is a sequence $\pi=(\pi_1, \dots, \pi_m)$ of positive integers whose non-decreasing rearrangement $(\lambda_1, \dots,…
We study the existence and construction of sparse supports for hypergraphs derived from subgraphs of a graph $G$. For a hypergraph $(X,\mathcal{H})$, a support $Q$ is a graph on $X$ s.t. $Q[H]$, the graph induced on vertices in $H$ is…
Let $X$ be a curve over $\F_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of…
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…
Let $\mathcal{G}$ be the set of all the planar embeddings of a (not necessarily connected) $n$-vertex graph $G$. We present a bijection $\Phi$ from $\mathcal{G}$ to the natural numbers in the interval $[0 \dots |\mathcal{G}| - 1]$. Given a…
The chromatic symmetric function $X_H$ of a hypergraph $H$ is the generating function for all colorings of $H$ so that no edge is monochromatic. When $H$ is an ordinary graph, it is known that $X_H$ is positive in the fundamental…
We introduce type C parking functions, encoded as vertically labelled lattice paths and endowed with a statistic dinv'. We define a bijection from type C parking functions to regions of the Shi arrangement of type C, encoded as diagonally…
We introduce the 2-sorted counting logic $GC^k$ that expresses properties of hypergraphs. This logic has available k variables to address hyperedges, an unbounded number of variables to address vertices, and atomic formulas E(e,v) to…
In parking problems, a given number of cars enter a one-way street sequentially, and try to park according to a specified preferred spot in the street. Various models are possible depending on the chosen rule for collisions, when two cars…