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We provide an explicit formulation for the solution to the Catalan's triangle system using Catalan's trapezoids and a specified boundary condition. Additionally, we study this system with various boundary conditions obtained by utilizing different types of Fock spaces. |
For a graph $G=(V,E)$, $k\in \mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as \[ {\bf Z}(G;k,w):=\sum_{\phi:V\to [k]}\prod_{\substack{uv\in E \\ \phi (u)=\phi |
Random 2-cell embeddings of a given graph $G$ are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, $\mathbb{E}[F_G]$, of such an embedding which is equivalent to studying its average genus. |
In this note we study the logarithmic derivation module of a non-free arrangement. We prove a generalized addition theorem for all arrangements. |
It is proved that for every $d\ge 2$ such that $d-1$ divides $q-1$, where $q$ is a power of 2, there exists a Denniston maximal arc $A$ of degree $d$ in $\PG(2,q)$, being invariant under a cyclic linear group that fixes one point of $A$ and acts regularly on the set of the remaining points of ${A}$. Two alternative pr... |
The transmission ${\rm Tr}_G (v)$ of a vertex $v$ of a connected graph $G$ is the sum of distances between $v$ and all other vertices in $G$. |
W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left\{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\right\} $ by showing that for any finite coloring of $X_{k}$ there exists a sequence such that the Gower subspace generated by that sequence is monochromatic. For $k=1,$ this immediately gives ... |
Let $G=(V,E)$ be a simple connected graph. A set of vertices $S\subseteq V$ is said to be a dominating set if for any vertex in $V\setminus S$ is adjacent to at least one vertex in $S$. |
We are concerned with the invariants of Steiner chains consisting of four circles. In particular, we compute the invariant moments of curvatures in Steiner 4-chains and give two applications of the obtained formulas. |
We introduce and study affine analogues of the fixed-point-free (FPF) involution Stanley symmetric functions of Hamaker, Marberg, and Pawlowski. Our methods use the theory of quasiparabolic sets introduced by Rains and Vazirani, and we prove that the subset of FPF-involutions is a quasiparabolic set for the affine sym... |
A bipartite graph G is known to be Pfaffian if and only if it does not contain an even subdivision H of $K_{3,3}$ such that $G - VH$ contains a 1-factor. However a general characterisation of Pfaffian graphs in terms of forbidden subgraphs is currently not known. |
Homomorphism duality pairs play crucial role in the theory of relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. |
The classical hook length formula of enumerative combinatorics expresses the number of standard Young tableaux of a given partition shape as a single fraction. In recent years, two generalizations of this formula have emerged: one by Pak and Postnikov, replacing the number by a (rational) generating function, and one ... |
If ${\cal F}$ is a set of subgraphs $F$ of a finite graph $E$ we define a graph-counting polynomial $$ p_{\cal F}(z)=\sum_{F\in{\cal F}}z^{|F|} $$ In the present note we consider oriented graphs and discuss some cases where ${\cal F}$ consists of unbranched subgraphs $E$. We find several situations where something can... |
The dichromatic number of a graph $G$ is the maximum integer $k$ such that there exists an orientation of the edges of $G$ such that for every partition of the vertices into fewer than $k$ parts, at least one of the parts must contain a directed cycle under this orientation. In 1979, Erdős and Neumann-Lara conjectured... |
In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not {\em who} wins but rather {\em how fast} can one win. |
The object of this paper is to propose and prove a new generalization of the Andrews-Gordon identities, extending a recent result of Garrett, Ismail and Stanton. We also give a combinatorial discussion of the finite form of their result, which appeared in the work of Andrews, Knopfmacher, and Paule. |
A code is called propelinear if its automorphism group contains a subgroup that acts regularly on its codewords, which is called a propelinear structure on the code. In the paper a classification of the propelinear structures on the Nordstrom-Robinson code is obtained and the question of extension of these structures ... |
In this paper, we prove that the largest pure partial plane of order 6 has size 25. At the same time, we classify all pure partial planes of order 6 and size 25 up to isomorphism. |
Robin Forman's highly influential 2002 paper A User's Guide to Discrete Morse Theory presents an overview of the subject in a very readable manner. As a proof of concept, the author determines the topology (homotopy type) of the abstract simplicial complex of disconnected graphs of order $n$ (which was previou... |
A conjecture of Graver from 1991 states that the generic $3$-dimensional rigidity matroid is the unique maximal abstract $3$-rigidity matroid with respect to the weak order on matroids. Based on a close similarity between the generic $d$-dimensional rigidity matroid and the generic $C_{d-2}^{d-1}$-cofactor matroid fro... |
Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32\Delta+1$ colors are sufficient to square color every planar graph of maximum degree $\Delta$. |
The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity $\lambda_k(G)$, recently. |
We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions. |
Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every $d$-degenerate graph is also weakly $d$-degenerate. |
In this paper we study the algebra of graph invariants, focusing mainly on the invariants of simple graphs. <br>All other invariants, such as sorted eigenvalues, degree sequences and canonical permutations, belong to this algebra. |
A graph $G=(V,E)$ is a $k$-critical graph if $G$ is not $(k -1)$-colorable but $G-e$ is $(k-1)$-colorable for every $e\in E(G)$. In this paper, we construct a family of 4-critical planar graphs with $n$ vertices and $\frac{7n-13}{3}$ edges. |
We study the structure and enumeration of the final two 2x4 permutation classes, completing a research program that has spanned almost two decades. For both classes, careful structural analysis produces a complicated functional equation. |
If $L$ is a list assignment of $r$ colors to each vertex of an $n$-vertex graph $G$, then an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/r\rceil$ times. A graph is equitably $r$-choosable if it has an equitable $L$-coloring for e... |
We introduce the Ungarian Markov chain ${\bf U}_L$ associated to a finite lattice $L$. The states of this Markov chain are the elements of $L$. |
Webs give a diagrammatic calculus for spaces of tensor invariants. We introduce hourglass plabic graphs as a new avatar of webs, and use these to give the first rotation-invariant $U_q(\mathfrak{sl}_4)$-web basis, a long-sought object. |
Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. |
For the Erdős-Rényi random graph G(n,p), we give a precise asymptotic formula for the size of a largest vertex subset in G(n,p) that induces a subgraph with average degree at most t, provided that p = p(n) is not too small and t = t(n) is not too large. In the case of fixed t and p, we find that this value is asymptot... |
We demonstrate the use of several tools from Algebraic Combinatorics such as Young tableaux, symmetry operators, the Littlewood-Richardson rule and discrete Fourier transforms of symmetric groups in investigations of algebraic curvature tensors. |
We prove that the edge-end space of an infinite graph is metrizable if and only if it is first-countable. This strengthens a recent result by Aurichi, Magalhaes Jr. |
We describe two similar but independently-coded computations used to construct a complete catalogue of the transitive groups of degree less than $48$, thereby verifying, unifying and extending the catalogues previously available. From this list, we construct all the vertex-transitive graphs of order less than $48$. |
Let $P_r(n)$ be the set of partitions of n with non negative rth differences. Let $\lambda$ be a partition chosen uniformly at random among the set $P_r(n)$. |
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$. |
Building off recent work of Garg and Peng, we continue the investigation into classical and consecutive pattern avoidance in rooted forests, resolving some of their conjectures and questions and proving generalizations whenever possible. Through extensions of the forest Simion-Schmidt bijection introduced by Anders an... |
Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e. any graphon determined by finitely many graph densities, has a simple structure. |
Beginning with the concepts of orientation for a 3-hypergraph and transitivity for an oriented 3-hypergraph, it is natural to study the class of comparability 3-hypergraphs (those that can be transitively oriented). In this work we show three different behaviors in respect to the relationship between the chromatic num... |
We prove that for any pair of constants $\epsilon>0$ and $\Delta$ and for $n$ sufficiently large, every family of trees of orders at most $n$, maximum degrees at most $\Delta$, and with at most $\binom{n}{2}$ edges in total packs into $K_{(1+\epsilon)n}$. This implies asymptotic versions of the Tree Packing Conject... |
For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set $\{1,\ldots,N\}$ contains a monochromatic copy of $G$ whose vertices follow the prescri... |
We say that a family of permutations $t$-shatters a set if it induces at least $t$ distinct permutations on that set. What is the minimum number $f_k(n,t)$ of permutations of $\{1, \dots, n\}$ that $t$-shatter all subsets of size $k$? |
For a graph $G$, let $\chi (G)$ denote the chromatic number. In graph theory, the following famous conjecture posed by Hedetniemi has been studied: For two graphs $G$ and $H$, $\chi (G\times H)=\min\{\chi (G),\chi (H)\}$, where $G \times H$ is the tensor product of $G$ and $H$. |
Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $ r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $ \binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. |
We improve upper bounds on the chromatic number proven independently in \cite{reedNote} and \cite{ingo}. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. |
The famous Goldbach conjecture remains open for nearly three centuries. Recently Goldbach graphs are introduced to relate the problem with the literature of Graph Theory. |
Define an expansion poset to be the poset of monomials of a cluster variable attached to an arc in a polygon, where each monomial is represented by the corresponding combinatorial object from some fixed combinatorial cluster expansion formula. We introduce an involution on several of the interrelated combinatorial obj... |
Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. |
The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of ${\rm PG}(2,q)$ remains an arc in the Hall plane obtained by derivation. |
Given a graph $G$ and an $r$-edge-colouring $\chi$ on $E(G)$, a Hamilton cycle $H\subset G$ is said to have $t$ colour-bias if $H$ contains $n/r+t$ edges of the same colour in $\chi$. Freschi, Hyde, Lada and Treglown showed every $r$-coloured graph $G$ on $n$ vertices with $\delta(G)\geq(r+1)n/2r+t$ contains a Hamilto... |
Richard P. Stanley defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order quasisymmetric functions, which are analogs of the chromatic symmetric functions, distingui... |
Zero forcing is a combinatorial game played on a graph with the ultimate goal of changing the colour of all the vertices at minimal cost. Originally this game was conceived as a one player game, but later a two-player version was devised in-conjunction with studies on the inertia of a graph, and has become known as th... |
Cover-Incomparability graphs (C-I graphs) are an interesting class of graphs from posets. A C-I graph is a graph from a poset $P=(V,\le)$ with vertex set $V$, and the edge-set is the union of edge sets of the cover graph and the incomparability graph of the poset. |
Let $G$ be a $\{C_4, C_5\}$-free planar graph with a list assignment $L$. Suppose a preferred color is given for some of the vertices. |
The Bruhat order on permutations arises out of the study of Schubert varieties in Grassmannians and flag varieties, which have been important for over 100 years. The purpose of this paper is to study variations on this theme related to subvarieties of the spanning line configurations $X_{n,k}$ as defined by Pawlowski ... |
We give an overview about the power product expansion of the exponential series and derive some q-analogs |
We prove that $\chi(G) \leq \max {\omega(G), \Delta_2(G), (5/6)(\Delta(G) + 1)}$ for every graph $G$ with $\Delta(G) \geq 3$. Here $\Delta_2$ is the parameter introduced by Stacho that gives the largest degree that a vertex $v$ can have subject to the condition that $v$ is adjacent to a vertex whose degree is at least... |
We consider a reconfiguration version of the homomorphism problem ${\rm Hom}_\mathbb{M}(N)$ for binary matroids $N$. This reconfiguration problem, ${\rm Recol}_\mathbb{M}(N)$, asks, for two homomorphisms $\phi$ and $\psi$ of a matroid $M$ to $N$, if there is a path of homomorphism from $\phi$ to $\psi$ such that conse... |
We call a graph $G$ an $(r,t)$-Ruzsa-Szemerédi graph if its edge set can be partitioned into $t$ edge-disjoint induced matchings, each of size $r$. These graphs were introduced in 1978 and has been extensively studied since then. |
We show that there exists an absolute constant $C>0$ such that any family $\mathcal{F}\subset \{0,1\}^n$ of size at least $Cn^3$ has dual VC-dimension at least 3. Equivalently, every family of size at least $Cn^3$ contains three sets such that all eight regions of their Venn diagram are non-empty. |
Let $t_{i,j}$ be the coefficient of $x^iy^j$ in the Tutte polynomial $T(G;x,y)$ of a connected bridgeless and loopless graph $G$ with order $n$ and size $m$. It is trivial that $t_{0,m-n+1}=1$ and $t_{n-1,0}=1$. |
We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree $d$. |
In graph theory, the Szemerédi regularity lemma gives a decomposition of the indicator function for any graph $G$ into a structured component, a uniform part, and a small error. This result, in conjunction with a counting lemma that guarantees many copies of a subgraph $H$ provided a copy of $H$ appears in the structu... |
Given $2k-1$ convex sets in $R^2$ such that no point of the plane is covered by more than $k$ of the sets, is it true that there are two among the convex sets whose union contains all $k$-covered points of the plane? This question due to Gy. |
There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs an... |
Let $G=(V,E)$ be a matching-covered graph, denote by $P$ its perfect matching polytope, and by $L$ the integer lattice generated by the integral points in $P$. In this paper, we give polyhedral proofs for two difficult results established by Lovász (1987), and by Carvalho, Lucchesi, and Murty (2002) in a series of thr... |
We show how a simplicial complex arising from the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations of string theory is the Whitehouse complex. Using discrete Morse theory, we give an elementary proof that the Whitehouse complex $\Delta_n$ is homotopy equivalent to a wedge of $(n-2)! |
A finite graph $\G$ is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all fin... |
We prove that the set of permutations sorted by a stack of depth $t \geq 3$ and an infinite stack in series has infinite basis, by constructing an infinite antichain. This answers an open question on identifying the point at which, in a sorting process with two stacks in series, the basis changes from finite to infini... |
In this paper, we justify by the use of Enumerative Combinatorics, that the results obtained in \cite{Alarcon1}, where is analysed the complex dynamics of an epidemic model to identify the nodes that contribute the most to the propagation process and because of that are good candidates to be controlled in the network i... |
The Brion morphism maps a generalized permutahedron to a collection of posets associated to its vertices. We compute this map explicitly for the Hopf monoids of permutahedra, associahedra, and orbit polytopes, and we explore the dual Brion map of the primitive Lie monoids associated to these three Hopf monoids. |
We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. |
Let $G$ be a connected graph with maximum degree $\Delta \ge 3$. We investigate the upper bound for the chromatic number $\chi_\gamma(G)$ of the power graph $G^\gamma$. |
Chao and Yu introduced an entropy method for hypergraph Turán problems, and used it to show that the family of $\lfloor k/2\rfloor$ $k$-uniform tents have Turán density $k! /k^k$. |
We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formu... |
We study a super group of the group of Riordan arrays, where the elements of the group are given by a triple of power series. We show that certain subsets are subgroups, and we identify a normal subgroup whose cosets correspond to Riordan arrays. |
In their famous 1974 paper introducing the local lemma, Erdős and Lovász posed a question-later referred by Erdős as one of his three favorite open problems: What is the minimum number of edges in an $r$-uniform, intersecting hypergraph with cover number $r$? This question was solved up to a constant factor in Kahn... |
We present a broader framework for the Cauchy identity derived from the determinant expansion of collocation matrices. This approach yields an infinite family of identities, where the original Cauchy identity stands as a particular case. |
Given a digraph $D=(V,A)$ with vertex-set $V=\{1,\ldots,n\}$ and arc-set $A$, we denote by $Q(D)$ the set of all real $n\times n$ matrices $B=[b_{u,w}]$ with $b_{u,u}\not=0$ for all $u\in V$, $b_{u,w} \not= 0$ if $u\not=w$ and there is an arc from $u$ to $w$, and <br>$b_{u,w}=0$ if $u\not=w$ and there is no arc from $u... |
The dichromatic number $\vec{\chi}(G)$ of a digraph $G$ is the least integer $k$ such that $G$ can be partitioned into $k$ acyclic digraphs. A digraph is $k$-dicritical if $\vec{\chi}(G) = k$ and each proper subgraph $H$ of $G$ satisfies $\vec{\chi}(H) \leq k-1$. |
Let $n \equiv 0\, (\, \text{mod } 3\,)$ and $H_{n, n/3}^2$ be the 3-graph of order $n$, whose vertex set is partitioned into two sets $S$ and $T$ of size $\frac{1}{3}n+1$ and $\frac{2}{3}n -1$, respectively, and whose edge set consists of all triples with at least $2$ vertices in $T$. Suppose that $n$ is sufficiently ... |
In this paper, we give necessary and sufficient conditions for a real symmetric matrix, and in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank one matrix. As a consequence, we present a short proof of the... |
Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for $\overline{p}(n)$ and derived a number of congruences for $\overline{p}(n)$ modulo $4$, $8$ and $64$ including $\overlin... |
This survey paper contains a tutorial introduction to distance-regular graphs, with an emphasis on the subconstituent algebra and the $Q$-polynomial property. |
We provide a simple proof of a curious inequality for the binary entropy function, an inequality that has been used in two different contexts. In the 1980's, Boppana used this entropy inequality to prove lower bounds on Boolean formulas. |
In the spirit of Bar Natan's construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers $\vec{x}=(x_1,...,x_n)$, we construct a commutative diagram in the shape of the Bruhat order on $S_n$ whose nodes are colored smoothings of the $2$-strand... |
In the paper we study choice functions on posets satisfying the conditions of heredity and outcast. For every well-ordered sequence of elements of poset, we define the corresponding `elementary' choice function. |
The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $p$ such that vertices of $G$ can be partitioned into disjoint classes $X_{1}, ..., X_{p}$ where vertices in $X_{i}$ have pairwise distance greater than $i$. For $k < t$ we study the packing chromatic number of infinite distance gr... |
Let $G$ be a simple graph with vertex set $V$ and edge set $E$, and let $S \subseteq V$. The \emph{open neighborhood} of $v \in V$, $N(v)$, is the set of vertices adjacent to $v$; the \emph{closed neighborhood} is given by $N[v] = N(v) \cup \{v\}$. |
The bipartite Ramsey number $B(n_1,n_2,\ldots,n_t)$ is the least positive integer $b$ such that, any coloring of the edges of $K_{b,b}$ with $t$ colors will result in a monochromatic copy of $K_{n_i,n_i}$ in the $i-$th color, for some $i$, $1\leq i\leq t$. In this paper we obtain the exact values of bipartite Ramsey n... |
A boolean term order is a total order on subsets of [n]={1,...,n} such that \emptyset < alpha for all nonempty alpha contained in [n], and alpha < beta implies alpha \cup gamma < beta \cup gamma for all gamma which do not intersect alpha or beta. Boolean term orders arise in several different areas of mathema... |
We investigate finite 3-nets embedded in a projective plane over a (finite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises concurrent lines, and irregular otherwise. |
For non-negative integers~$k$, we consider graphs in which every vertex has exactly $k$ vertices at distance~$2$, i.e., graphs whose distance-$2$ graphs are $k$-regular. We call such graphs $k$-metamour-regular motivated by the terminology in polyamory. |
We say that a graph $G$ has an {\em odd $K_4$-subdivision} if some subgraph of $G$ is isomorphic to a $K_4$-subdivision and whose faces are all odd holes of $G$. For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of graphs which have girth $2\ell+1$ and have no odd hole with length greater than $2\e... |
In this paper, we consider a randomized greedy algorithm for independent sets in $r$-uniform $d$-regular hypergraphs $G$ on $n$ vertices with girth $g$. By analyzing the expected size of the independent sets generated by this algorithm, we show that $\alpha(G)\geq (f(d,r)-\epsilon(g,d,r))n$, where $\epsilon(g,d,r)$ co... |
We show that there is a positive constant $c$ such that any graph on vertex set $[n]$ with at most $c n^2/k^2 \log k$ edges contains an independent set of order $k$ whose vertices form an arithmetic progression. We also present applications of this result to several questions in Ramsey theory. |
In this paper we present results concerning the stabilizer $G_f$ in $\mathrm{GL}(2,q^n)$ of the subspace $U_f=\{(x,f(x))\colon x\in\mathbb F_{q^n}[x]\}$, $f(x)$ a scattered linearized polynomial in $\mathbb F_{q^n}[x]$. Each $G_f$ contains the $q-1$ maps $(x,y)\mapsto(ax,ay)$, $a\in\mathbb F_{q}^*$. |
Let ${\rm gp}(G)$ be the general position number of a graph $G$. It is proved that ${\rm gp}(G-x)\leq 2{\rm gp}(G)$ holds for any vertex $x$ of a connected graph $G$ and that if $x$ lies in some ${\rm gp}$-set of $G$, then ${\rm gp}(G) - 1 \le {\rm gp}(G-x)$. |
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