Let P_G(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing P_G(4) over all planar graphs G. Since then, motivated by a variety of applications, much research was done on minimizing or maximizing P_G(q) over various families of graphs.
In this paper, we study an old problem of Linial and Wilf, to find the graphs with n vertices and m edges which maximize the number of q-colorings. We provide the first approach which enables one to solve this problem for many nontrivial ranges of parameters. Using our machinery, we show that for each q >= 4 and sufficiently large m < κ_q n^2 where κ_q is approximately 1/(q log q), the extremal graphs are complete bipartite graphs minus the edges of a star, plus isolated vertices. Moreover, for q = 3, we establish the structure of optimal graphs for all large m <= n^2/4, confirming (in a stronger form) a conjecture of Lazebnik from 1989.
We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and Radcliffe on the maximum and minimum number of existence homomorphisms from a tree to a completely looped graph on $q$ vertices.
For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex graph with minimum degree $δ$ maximizes the number of $H$-colorings? We show that for non-regular $H$ and sufficiently large $n$, the complete bipartite graph $K_{δ,n-δ}$ is the unique maximizer. As a corollary, for non-regular $H$ and sufficiently large $n$ the graph $K_{k,n-k}$ is the unique $k$-connected graph that maximizes the number of $H$-colorings among …
Let $\mathcal{C}_4(n)$ be the family of all connected $4$-chromatic graphs of order $n$. Given an integer $x\geq 4$, we consider the problem of finding the maximum number of $x$-colorings of a graph in $\mathcal{C}_4(n)$. It was conjectured that the maximum number of $x$-colorings is equal to $(x)_{\downarrow 4}(x-1)^{n-4}$ and the extremal graphs are those which have clique number $4$ and size $n+2$. In this article, we reduce this problem to a \textit{finite} family of graphs. We show that there exist a finite family $\mathcal{F}$ of connected $4$-chromatic graphs such that if the number of …
The number of proper $q$-colorings of a graph $G$, denoted by $P_G(q)$, is an important graph parameter that plays fundamental role in graph theory, computational complexity theory and other related fields. We study an old problem of Linial and Wilf to find the graphs with $n$ vertices and $m$ edges which maximize this parameter. This problem has attracted much research interest in recent years, however little is known for general $m,n,q$. Using analytic and combinatorial methods, we characterize the asymptotic structure of extremal graphs for fixed edge density and $q$. Moreover, we disprove …
For graphs $G$ and $H$, a {\em homomorphism} from $G$ to $H$, or {\em $H$-coloring} of $G$, is an adjacency preserving map from the vertex set of $G$ to the vertex set of $H$. Writing ${\rm hom}(G,H)$ for the number of $H$-colorings admitted by $G$, we conjecture that for any simple finite graph $H$ (perhaps with loops) and any simple finite $n$-vertex, $d$-regular, loopless graph $G$ we have $$ {\rm hom}(G,H) \leq \max{{\rm hom}(K_{d,d},H)^{\frac{n}{2d}}, {\rm hom}(K_{d+1},H)^{\frac{n}{d+1}}} $$ where $K_{d,d}$ is the complete bipartite graph with $d$ vertices in each partition class, and $K_…
Let $P_G(k)$ be the number of proper $k$-colorings of a finite simple graph $G$. Tomescu's conjecture, which was recently solved by Fox, He, and Manners, states that $P_G(k) \le k!(k-1)^{n-k}$ for all connected graphs $G$ on $n$ vertices with chromatic number $k\geq 4$. In this paper, we study the same problem with the additional constraint that $G$ is $\ell$-connected. For $2$-connected graphs $G$, we prove a tight bound \[ P_G(k) \le (k-1)!((k-1)^{n-k+1} + (-1)^{n-k}), \] and show that equality is only achieved if $G$ is a $k$-clique with an ear attached. For $\ell \ge 3$, we prove an asympt…
A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives a distinct color. In this paper we consider the natural extremal problem of maximizing and minimizing the number of rainbow spanning trees in a graph $G$. Such a question clearly needs restrictions on the colorings to be meaningful. For edge-colorings using $n-1$ colors and without rainbow cycles, known in the literature as JL-colorings, there turns out to be a particularly nice way of counting the rainbow spanning trees and we solve this problem completely for JL-colored complete graphs $K_n$ and comp…
We consider the graph coloring game, a game in which two players take turns properly coloring the vertices of a graph, with one player attempting to complete a proper coloring, and the other player attempting to prevent a proper coloring. We show that if a graph $G$ has a proper coloring in which the game coloring number of each bicolored subgraph is bounded, then the game chromatic number of $G$ is bounded. As a corollary to this result, we show that for two graphs $G_1$ and $G_2$ with bounded game coloring number, the Cartesian product $G_1 \square G_2$ has bounded game chromatic number, ans…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle. Many known upper bounds for the list-chromatic number extend to the DP-chromatic number, but not all of them do. In this note we describe some unusual properties of DP-coloring that set it aside from list coloring. In particular, we give an example of a planar bipartite graph with DP-chromatic number $4$ and prove that the edge-DP-chromatic number of a $d$-regular graph with $d\geq 2$ is always at least $d+1$.
A $k$-ranking is a vertex $k$-coloring such that if two vertices have the same color any path connecting them contains a vertex of larger color. The rank number of a graph is smallest $k$ such that $G$ has a $k$-ranking. For certain graphs $G$ we consider the maximum number of edges that may be added to $G$ without changing the rank number. Here we investigate the problem for $G=P_{2^{k-1}}$, $C_{2^{k}}$, $K_{m_{1},m_{2},\dots,m_{t}}$, and the union of two copies of $K_{n}$ joined by a single edge. In addition to determining the maximum number of edges that may be added to $G$ without changing…
A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring. In this article, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitely the total dominator chromatic number of the middle graph of several known families of graphs.
Let $G$ be a simple graph of order $n$. A majority dominator coloring of a graph $G$ is proper coloring in which each vertex of the graph dominates at least half of one color class. The majority dominator chromatic number $χ_{md}(G)$ is the minimum number of color classes in a majority dominator coloring of $G$. In this paper we study properties of the majority dominator coloring of a graph. We obtain tight upper and lower bounds in terms of chromatic number, dominator chromatic number, maximum degree, domination and independence number. We also study majority dominator coloring number of sele…
We consider the \emph{connected game coloring number} of a graph, introduced by Charpentier et al. as a game theoretic graph parameter that measures the degeneracy of a graph with respect to a certain two-player game played with an uncooperative adversary. We consider the connected game coloring number of graphs of bounded treedepth and of $k$-trees. In particular, we show that there exists an outerplanar $2$-tree with connected game coloring number of $5$, which answers a question from [C. Charpentier, H. Hocquard, E. Sopena, and X. Zhu. A connected version of the graph coloring game. \textit…
The game coloring number gcol($G$) of a graph $G$ is a two player competitive variant of the coloring number. We introduce the preordered game coloring number to study the consequences of either player skipping any number of turns. In particular, we show that neither player can improve their performance by doing so. We use this result to show that for any induced subgraph $H \subset G$, $|G-H| = k$ implies the tight bound gcol($H$) $\leq$ gcol($G$) $-$ $2k$ and if $d_G(x)$ $\leq$ gcol($G-x$), then gcol($G-x$) $\leq$ gcol($G$) $-$ $1$.
We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such a coloring. We show that this notion generalizes the dominator and total dominator chromatic numbers, and provide some general bounds and algorithmic results. We also investigate the specific cases where $P$ is that the subset contains at least one edge or that the subset is connected.
A vertex coloring of a graph $G$ is called distinguishing (or symmetry breaking) if no non-identity automorphism of $G$ preserves it, and the distinguishing number, shown by $D(G)$, is the smallest number of colors required for such a coloring. This paper is about counting non-equivalent distinguishing colorings of graphs with $k$ colors. A parameter, namely $Φ_k (G)$, which is the number of non-equivalent distinguishing colorings of a graph $G$ with at most $k$ colors, is shown here to have an application in calculating the distinguishing number of the lexicographic product and the $X$-join o…
A well-known result of Alon shows that the coloring number of a graph is bounded by a function of its choosability. We explore this relationship in a more general setting with relaxed assumptions on color classes, encoded by a graph parameter.
The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia--like bound and a ratio--like bound are shown. A Sandwich Theorem involving the clique number, the vector chromatic number and the coloring number is proved, as well as a lower bound for the vector chromatic number in terms of the smallest and the largest eigenvalue of the normalized Laplacian. In addition, spectral partition numbers are studied in relation to the coloring numb…
DP-coloring is a generalization of list coloring introduced recently by Dvořák and Postle. We prove that for every $n$-vertex graph $G$ whose chromatic number $χ(G)$ is "close" to $n$, the DP-chromatic number of $G$ equals $χ(G)$. "Close" here means $χ(G)\geq n-O(\sqrt{n})$, and we also show that this lower bound is best possible (up to the constant factor in front of $\sqrt{n}$), in contrast to the case of list coloring.
In this paper, we prove that a graph $G$ with no $K_{s,s}$-subgraph and twin-width $d$ has $r$-admissibility and $r$-coloring numbers bounded from above by an exponential function of $r$ and that we can construct graphs achieving such a dependency in $r$.
We introduce a new concept in graph coloring motivated by the popular Sudoku puzzle. Let $G=(V,E)$ be a graph of order $n$ with chromatic number $χ(G)=k$ and let $S\subseteq V.$ Let $\mathscr C_0$ be a $k$-coloring of the induced subgraph $G[S].$ The coloring $\mathscr C_0$ is called an extendable coloring if $\mathscr C_0$ can be extended to a $k$-coloring of $G.$ We say that $\mathscr C_0$ is a Sudoku coloring of $G$ if $\mathscr C_0$ can be uniquely extended to a $k$-coloring of $G.$ The smallest order of such an induced subgraph $G[S]$ of $G$ which admits a Sudoku coloring is called the Su…
Let G be a simple graph. A coloring of vertices of G is called (i) a 2-proper coloring if vertices at distance 2 receive distinct colors; (ii) an injective coloring if vertices possessing a common neighbor receive distinct colors; (iii) a square coloring if vertices at distance at most 2 receive distinct colors. In this paper, we study inequalities of Nordhaus-Guddam type for the 2-proper chromatic number, the injective chromatic number, and the square chromatic number.
An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$ with $k\le 2^{n/4300}$, the number of Gallai colorings of $K_n$ that use at most $k$ given colors is $(\binom{k}{2}+o_n(1))\,2^{\binom{n}{2}}$. Our result is asymptotically best possible and implies that, for those $k$, almost all Gallai $k$-colorings use only two colors. However, this is not true for $k \ge Ω(2^{2n})$.
Weak coloring numbers generalize the notion of degeneracy of a graph. They were introduced by Kierstead \& Yang in the context of games on graphs. Recently, several connections have been uncovered between weak coloring numbers and various parameters studied in graph minor theory and its generalizations. In this note, we show that for every fixed $k\geq1$, the maximum $r$-th weak coloring number of a graph with simple treewidth $k$ is $Θ(r^{k-1}\log r)$. As a corollary, we improve the lower bound on the maximum $r$-th weak coloring number of planar graphs from $Ω(r^2)$ to $Ω(r^2\log r)$, and we…
We establish new lower bounds for $28$ classical two and three color Ramsey numbers, and describe the heuristic search procedures we used. Several of the new three color bounds are derived from the two color constructions; specifically, we were able to use $(5,k)$-colorings to obtain new $(3,3,k)$-colorings, and $(7,k)$-colorings to obtain new $(3,4,k)$-colorings. Some of the other new constructions in the paper are derived from two well-known colorings: the Paley coloring of $K_{101}$ and the cubic coloring of $K_{127}$.
Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring such that every color class is dominated with at least one vertex. The dominator chromatic number of corona products and of edge corona products is determined. Sharp lower and upper bounds are given for the dominated chromatic number of edge corona products. The dominator chromatic number of hierarchical products is bounded from above and the dominated chro…
Let $Q_d$ be the $d$-dimensional hypercube and $N=2^d$. We prove that the number of (proper) 4-colorings of $Q_d$ is asymptotically \[6e2^N,\] as was conjectured by Engbers and Galvin in 2012. The proof uses a combination of information theory (entropy) and isoperimetric ideas originating in work of Sapozhenko in the 1980's.
We consider the chromatic numbers of unit-distance graphs of various annuli. In particular, we consider radial colorings, which are "nice" colorings, and completely determine the radial chromatic numbers of various annuli.
Given a proper coloring $f$ of a graph $G$, a b-vertex in $f$ is a vertex that is adjacent to every color class but its own. It is a b-coloring if every color class contains at least one b-vertex, and it is a fall-coloring if every vertex is a b-vertex. The b-chromatic number of $G$ is the maximum integer $b(G)$ for which $G$ has a b-coloring with $b(G)$ colors, while the fall-chromatic number and the fall-acromatic number of $G$ are, respectively, the minimum and maximum integers $f_1(G),f_2(G)$ for which $G$ has a fall-coloring. In this article, we explore the concepts of b-homomorphisms and…
A total dominator coloring of a graph $G$ is a proper coloring of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring of it. Here, we study the total dominator coloring on central graphs by giving some tight bounds for the total dominator chromatic number of the central of a graph, join of two graphs and Nordhaus-Gaddum-like relations. Also we will calculate the total dominator chromatic number of the central of a path, a cycle, a wheel, a comp…
We prove that the two-colouring number of any planar graph is at most 8. This resolves a question of Kierstead et al. [SIAM J. Discrete Math.~23 (2009), 1548--1560]. The result is optimal.
Let G be a simple connected graph. Then chi L(G) and chi D(G) will denote the locating chromatic number and the distinguishing chromatic number of G, respectively. In this paper, we investigate a comparison between chi L(G) and chi D(G). In fact, we prove that chi D(G) \leq chi L(G). Moreover, we determine some types of graphs whose locating and distinguishing chromatic numbers are equal. Specially, we characteristic all graph G with the property that chi D(G)= chi L(G) = 3.
A family of sets is called star-shaped if all the members of the family have a point in common. The main aim of this paper is to provide a negative answer to the following question raised by James Aisenberg et al [Short proofs of the kneser-Lovasz coloring principle, Information and Computation, 261:296-310, 2018.], for the case k=2.
The applications of graph coloring are diverse and many so lots of new types of coloring are being proposed and explored. Here we define a safe k-coloring, motivated by the application of coloring to secret sharing. Secret sharing is a way of securing a secret from a number of attackers by dividing it into parts and then distributing those parts to some persons, represented here by graph vertices. Parts of the secret are represented by colors which are then assigned to the vertices under certain conditions, making a coloring safe if a predetermined number of attackers cannot read the whole sec…
Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a structured class of colorings, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embed…
In this paper, we study the rainbow Erdős-Rothschild problem with respect to 3-term arithmetic progressions. We obtain the asymptotic number of $r$-colorings of $[n]$ without rainbow 3-term arithmetic progressions, and we show that the typical colorings with this property are 2-colorings. We also prove that $[n]$ attains the maximum number of rainbow 3-term arithmetic progression-free $r$-colorings among all subsets of $[n]$. Moreover, the exact number of rainbow 3-term arithmetic progression-free $r$-colorings of $\mathbb{Z}_p$ is obtained, where $p$ is any prime and $\mathbb{Z}_p$ is the cyc…
It is known that, for any $k$-list assignment $L$ of a graph $G$, the number of $L$-list colorings of $G$ is at least the number of the proper $k$-colorings of $G$ when $k>(m-1)/\ln(1+\sqrt{2})$. In this paper, we extend the Whitney's broken cycle theorem to $L$-colorings of signed graphs, by which we show that if $k> \binom{m}{3}+\binom{m}{4}+m-1$ then, for any $k$-assignment $L$, the number of $L$-colorings of a signed graph $Σ$ with $m$ edges is at least the number of the proper $k$-colorings of $Σ$. Further, if $L$ is $0$-free (resp., $0$-included) and $k$ is even (resp., odd), then the lo…
Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well studied notions of $(k,d)$-colorings and of the circular chromatic number $χ_c$ to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number $χ$. Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are si…
There are many concepts of signed graph coloring which are defined by assigning colors to the vertices of the graphs. These concepts usually differ in the number of self-inverse colors used. We introduce a unifying concept for this kind of coloring by assigning elements from symmetric sets to the vertices of the signed graphs. In the first part of the paper, we study colorings with elements from symmetric sets where the number of self-inverse elements is fixed. We prove a Brooks'-type theorem and upper bounds for the corresponding chromatic numbers in terms of the chromatic number of the under…
For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of $H$-colorings. In this paper, we prove several results in this area. First, we find a class of graphs ${\mathcal H}$ with the property that for each $H \in {\mathcal H}$, the $n$-vertex tree that minimizes the number of $H$-colorings is …
Let ${\rm col_g}(G)$ be the game coloring number of a given graph $G.$ Define the game coloring number of a family of graphs $\mathcal{H}$ as ${\rm col_g}(\mathcal{H}) := \max\{{\rm col_g}(G):G \in \mathcal{H}\}.$ Let $\mathcal{P}_k$ be the family of planar graphs of girth at least $k.$ We show that ${\rm col_g}(\mathcal{P}_7) \leq 5.$ This result extends a result about the coloring number by Wang and Zhang {WZ11} (${\rm col_g}(\mathcal{P}_8) \leq 5).$ We also show that these bounds are sharp by constructing a graph $G$ where $G \in {\rm col_g}(\mathcal{P}_k) \geq 5$ for each $k \leq 8$ such t…
The total chromatic number, $χ''(G)$ is the minimum number of colors which need to be assigned to obtain a total coloring of the graph $G$. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing that for any graph, $χ''(G) \leq Δ(G)+2 $, where $Δ(G)$ represents the maximum degree of $G$. In this paper we obtained the total chromatic number for some classes of four regular circulant graphs.
Let $\mathcal{C}_k(n)$ be the family of all connected $k$-chromatic graphs of order $n$. Given a natural number $x\geq k$, we consider the problem of finding the maximum number of $x$-colorings among graphs in $\mathcal{C}_k(n)$. When $k\leq 3$ the answer to this problem is known, and when $k\geq 4$ the problem is wide open. For $k\geq 4$ it was conjectured that the maximum number of $x$-colorings is $x(x-1)\cdots (x-k+1)\,x^{n-k}$. In this article, we prove this conjecture under the additional condition that the independence number of the graphs is at most $2$.
A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from $\{1, 2, \ldots, k\}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we consider two extremal problems related to Gallai-$k$-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle …
A star coloring of a graph $G$ is a proper vertex coloring such that the subgraph induced by any pair of color classes is a star forest. The star chromatic number of $G$ is the minimum number of colors needed to star color $G$. In this paper we determine the star-chromatic number of the splitting graphs of cycles of length $n$ with $n \equiv 1 \pmod 3$ and $n=5$, resolving an open question of Furnmańczyk, Kowsalya, and Vernold Vivin.
\qquad A \emph{coloring} of a digraph $D=(V,E)$ is a coloring of its vertices following the rule: Let $uv$ be an arc in $D$. If the tail $u$ is colored first, then the head $v$ should receive a color different from that of $u$. The \emph{dichromatic number} $χ_d(D)$ of $D$ is the minimum number of colors needed in a coloring of $D$. Besides obtaining many results and bounds for $χ_d(D)$ analogous to that of chromatic number of a graph, we prove $χ_d(D)=1$ if $D$ is acyclic. New notions of sequential colorings of graphs/digraphs are introduced. A characterization of acyclic digraph is obtained …
The online list coloring is a widely studied topic in graph theory. A graph $G$ is 2-paintable if we always have a strategy to complete a coloring in an online list coloring of $G$ in which each vertex has a color list of size 2. In this paper, we focus on the online list coloring game in which the number of colors is known in advance. We say that $G$ is $[2,t]$-paintable if we always have a strategy to complete a coloring in an online list coloring of $G$ in which we know that there are exactly $t$ colors in advance, and each vertex has a color list of size 2. Let $M(G)$ denote the maximum $t…
The Erdős-Gyárfás number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that all of its $p$-clique spans at least $q$ colors. In this paper we improve the best known upper bound on $f(n, p, q)$ for many fixed values of $p, q$ and large $n$. Our proof uses a randomized coloring process, which we analyze using the so-called differential equation method to establish dynamic concentration.
A proper vertex coloring $φ$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $φ^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of $G$, denoted $χ_o(G)$, is the odd chromatic number. Odd colorings were recently introduced in [M.~Petruševski, R.~Škrekovski: \textit{Colorings with neighborhood parity condition}]. Here we discuss various basic properties of this new graph parameter, characterize acyclic graphs and hypercubes in terms of odd chromatic number, establish several upper bounds in regard to d…
A \textit{locally identifying coloring} ($lid$-coloring) of a graph is a proper coloring such that the sets of colors appearing in the closed neighborhoods of any pair of adjacent vertices having distinct neighborhoods are distinct. Our goal is to study a \textit{relaxed locally identifying coloring} ($rlid$-coloring) of a graph that is similar to locally identifying coloring for which the coloring is not necessary proper.We denote by $χ_{rlid}(G)$ the minimum number of colors used in a relaxed locally identifying coloring of a graph $G$ In this paper, we prove that the problem of deciding tha…
