Alexander Dean, Sean English, Tongyun Huang
et al.
The firefighter problem with $k$ firefighters on an infinite graph $G$ is an iterative graph process, defined as follows: Suppose a fire breaks out at a given vertex $v\in V(G)$ on Turn 1. On each subsequent even turn, $k$ firefighters protect $k$ vertices that are not on fire, and on each subsequent odd turn, any vertex that is on fire spreads the fire to all adjacent unprotected vertices. The firefighters' goal is to eventually stop the spread of the fire. If there exists a strategy for $k$ firefighters to eventually stop the spread of the fire, then we say $G$ is $k$-containable. We consider the firefighter problem on the hexagonal grid, which is the graph whose vertices and edges are exactly the vertices and edges of a regular hexagonal tiling of the plane. It is not known if the hexagonal grid is $1$-containable. In arXiv:1305.7076 [math.CO], it was shown that if the firefighters have one firefighter per turn and one extra firefighter on two turns, the firefighters can contain the fire. We improve on this result by showing that even with only one extra firefighter on one turn, the firefighters can still contain the fire. In addition, we explore $k$-containability for birth sequence trees, which are infinite rooted trees that have the property that every vertex at the same level has the same degree. A birth sequence forest is an infinite forest, each component of which is a birth sequence tree. For birth sequence trees and forests, the fire always starts at the root of each tree. We provide a pseudopolynomial time algorithm to decide if all the vertices at a fixed level can be protected or not.
Jean-Christophe Aval, Adrien Boussicault, Bérénice Delcroix-Oger
et al.
We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differ- ential equation whose solution can be described combinatorially. This yield a new formula for the number of NATs. We also obtain q-versions of our formula. And we generalize NATs to higher dimension.
Edward Richmond, Vasu Tewari, Stephanie Van Willigenburg
The geometric Littlewood-Richardson (LR) rule is a combinatorial algorithm for computing LR coefficients derived from degenerating the Richardson variety into a union of Schubert varieties in the Grassmannian. Such rules were first given by Vakil and later generalized by Coskun. In this paper we give a noncommutative version of the geometric LR rule. As a consequence, we establish a geometric explanation for the positivity of noncommutative LR coefficients in certain cases.
We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, andsimplicial complexes. To this coloring there is an associated symmetric function in noncommuting variables for whichwe give a deletion-contraction formula. In the case of graphs our symmetric function in noncommuting variablesagrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloringis a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contractionlaw can be applied to scheduling problems.
In this work, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corre- sponding factorization theorems which in the classical case were established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind as well as Abel numbers. We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel.
AbstractLet$H$be a Hopf algebra. We consider$H$-equivariant modules over a Hopf module category${\mathcal{C}}$as modules over the smash extension${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the$H$-locally finite cohomologies of these objects. We also introduce relative$({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category${\mathcal{D}}$. These generalize relative$(A,H)$-Hopf modules over an$H$-comodule algebra$A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational$\text{Hom}$objects and higher derived functors of coinvariants.
Recently Naserasr, Sopena, and Zaslavsky [R. Naserasr, E. Sopena, T. Zaslavsky,Homomorphisms of signed graphs: An update, arXiv: 1909.05982v1 [math.CO] 12 Sep 2019.] published a report on closed walks in signed graphs. They gave a characterization of the sets of closed walks in a graph $G$ which corespond to the set of negative walks in some signed graph on $G$. In this note we show that their characterization is not valid and give a new characterization.
We study the mean-median map as a dynamical system on the space of finite multisets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive family of rational points, the local minima. By constructing a simpler map which represents the dynamics in such neighbourhoods, we extend the results of Cellarosi and Munday (arXiv:1408.3454v1 [math.CO]) by two orders of magnitude. Based on these computations, we conjecture that the Hausdorff dimension of the graph of the limit function of the multiset $[0,x,1]$ is greater than 1.
Let $S\neq\mathbb N$ be a numerical semigroup generated by $e$ elements. In his paper (A Circle-Of-Lights Algorithm for the "Money-Changing Problem", Amer. Math. Monthly 85 (1978), 562--565), H.~S.~Wilf raised the following question: Let $\Omega$ be the number of positive integers not contained in $S$ and $c-1$ the largest such element. Is it true that the fraction $\frac\Omega c$ of omitted numbers is at most $1-\frac1e$? Let $B\subseteq\mathbb N^{e-1}$ be the complement of an artinian $\mathbb N^{e-1}$-ideal. Following a concept of A.~Zhai (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1 [math.CO]) we relate Wilf's problem to a more general question about the weight distribution on $B$ with respect to a positive weight vector. An affirmative answer is given in special cases, similar to those considered by R.~Fr\"oberg, C.~Gottlieb, R.~H\"aggkvist (On numerical semigroups, Semigroup Forum, Vol.~35, Issue 1, 1986/1987, 63--83) for Wilf's question.
Let $S$ be a numerical semigroup of embedding dimension $e$ and conductor $c$. The question of Wilf is, if $\#(\mathbb N\setminus S)/c\leq e-1/e$. \noindent In (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1 [math.CO], 2011, Lemma 3), Zhai has shown an analogous inequality for the distribution of weights $x\cdot\gamma$, $x\in\mathbb N^d$, w.\,r. to a positive weight vector $\gamma$: \noindent Let $B\subseteq\mathbb N^d$ be finite and the complement of an $\mathbb N^d$-ideal. Denote by $\operatorname{mean}(B\cdot\gamma)$ the average weight of $B$. Then \[\operatorname{mean}(B\cdot\gamma)/\max(B\cdot\gamma)\leq d/d+1.\] $\bullet$ For the family $\Delta_n:=\{x\in\mathbb N^d|x\cdot\gamma<n+1\}$ of such sets we are able to show, that $\operatorname{mean}(\Delta_n\cdot\gamma)/\max(\Delta_n\cdot\gamma)$ converges to $d/d+1$, as $n$ goes to infinity. $\bullet$ Applying Zhai's Lemma 3 to the Hilbert function of a positively graded Artinian algebra yields a new class of numerical semigroups satisfying Wilf's inequality.
Marija Jeli'c, Duvsko Joji'c, Marinko Timotijevi'c
et al.
The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [n]$ of $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $\pi(K)\leq r$. Motivated by the problems of Tverberg-Van Kampen-Flores type, and inspired by the `constraint method' of Blagojevi\'{c}, Frick, and Ziegler, arXiv:1401.0690 [math.CO], we study the combinatorics of $r$-unavoidable complexes.
Stephan (Prove or Disprove 100 Conjectures from the OES, arXiv:math/0409509v4 [math.CO])enumerates a number of conjectures regarding integer sequences contained in Sloane's On-line Encyclopedia of Integer Sequences (N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences, published electronically at this https URL). In this paper, we prove two of these conjectures.
We prove that every digraph of circumference l has DAG‐width at most l. This is best possible and solves a recent conjecture from S. Kintali (ArXiv:1401.2662v1 [math.CO], January 2014).1 As a consequence of this result we deduce that the k‐linkage problem is polynomially solvable for every fixed k in the class of digraphs with bounded circumference. This answers a question posed in J. Bang‐Jensen, F. Havet, and A. K. Maia (Theor Comput Sci 562 (2014), 283–303). We also prove that the weak k‐linkage problem (where we ask for arc‐disjoint paths) is polynomially solvable for every fixed k in the class of digraphs with circumference 2 as well as for digraphs with a bounded number of disjoint cycles each of length at least 3. The case of bounded circumference digraphs is still open. Finally, we prove that the minimum spanning strong subdigraph problem is NP‐hard on digraphs of DAG‐width at most 5.
Following D.B. Karaguezian, V. Reiner, and M.L. Wachs (Matching Complexes, Bounded Degree Graph Complexes, and Weight Spaces of $GL$-Complexes, Journal of Algebra 2001) we study the connectivity degree and shellability of multiple chessboard complexes. Our central new results (Theorems 3.2 and 4.4) provide sharp connectivity bounds relevant to applications in Tverberg type problems where multiple points of the same color are permitted. These results also provide a foundation for the new results of Tverberg-van Kampen-Flores type, as announced in arXiv:1502.05290 [math.CO].
We use a recently introduced combinatorial object, the $\textit{interval-poset}$, to describe two bijections on intervals of the Tamari lattice. Both bijections give a combinatorial proof of some previously known results. The first one is an inner bijection between Tamari intervals that exchanges the $\textit{initial rise}$ and $\textit{lower contacts}$ statistics. Those were introduced by Bousquet-Mélou, Fusy, and Préville-Ratelle who proved they were symmetrically distributed but had no combinatorial explanation. The second bijection sends a Tamari interval to a closed flow of an ordered forest. These combinatorial objects were studied by Chapoton in the context of the Pre-Lie operad and the connection with the Tamari order was still unclear.
Dominique Bontemps, Stephane Boucheron, Elisabeth Gassiat
This paper sheds light on universal coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate. We prove that the auto-censuring (AC) code introduced by Bontemps (2011) is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy by Haussler and Opper (1997) and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on non-asymptotic bounds for maxima of samples from discrete distributions with finite and non-decreasing hazard rate.
Let G be a connected graph. The notion of rainbow connection number rc(G) of a graph G was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph G with radius r, $${rc(G)\leq r(r+2)}$$ and the bound is tight. In this paper, we show that for a connected graph G with radius r and center vertex u, if we let Dr = {u}, then G has r−1 connected dominating sets $${ D^{r-1}, D^{r-2},\ldots, D^{1}}$$ such that $${D^{r} \subset D^{r-1} \subset D^{r-2} \cdots\subset D^{1} \subset D^{0}=V(G)}$$ and $${rc(G)\leq \sum_{i=1}^{r} \max \{2i+1,b_i\}}$$, where bi is the number of bridges in E[Di, N(Di)] for $${1\leq i \leq r}$$. From the result, we can get that if $${b_i\leq 2i+1}$$ for all $${1\leq i\leq r}$$, then $${rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)}$$; if bi > 2i + 1 for all $${1\leq i\leq r}$$ , then $${rc(G)= \sum_{i=1}^{r}b_i}$$, the number of bridges of G. This generalizes the result of Basavaraju et al. In addition, an example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the radius of G, and another example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the number of bridges in G.