We discuss the problem of finding critical sets in graphs, a concept which has appeared in a number of guises in the combinatorics and graph theory literature. The case of the Sudoku graph receives particular attention, because critical sets correspond to minimal fair puzzles. We define four parameters associated with the sizes of extremal critical sets and (a) prove several general results about these parameters' properties, including their computational intractability, (b) compute their values exactly for some classes of graphs, (c) obtain bounds for generalized Sudoku graphs, and (d) offer a number of open questions regarding critical sets and the aforementioned parameters.
The Sudoku number has been defined under various names, indicating it is a natural concept. There are four variants of this parameter, that can be related to the maximum and minimum size of a critical set in a graph colouring problem. For each of these four related parameters, we present some simple characterizations of the graphs attaining the maximum possible values. As a main result, we answer a question by Cooper and Kirkpatrick, showing that there is monotone behaviour in the number of colours for only two of the four parameters. We investigate the monotone behaviour for the subgraph-orde…
This paper deals with a generalized Sudoku problem and investigates the unicity of a given solution. We introduce constraint sets, which is a generalization of the rows, columns and blocks of a classical Sudoku puzzle. The unicity property is characterized by three different properties. We describe unicity by permutations, by unicity cells and by rectangles. These terms are defined in this paper and are illustrated with examples. Throughout this paper we are not concerned with the existence of a solution.
We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-$n$ squares, single diagonal latin squares, double diagonal latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs.
In this paper we determine a class of critical sets in the abelian {2--group} that may be obtained from a greedy algorithm. These new critical sets are all 2--critical (each entry intersects an intercalate, a trade of size 4) and completes in a top down manner.
Let $γ(G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a \emph{critical vertex} of $G$ if $γ(G-v)=γ(G)-1$. A graph is called \emph{vertex-critical} if every vertex of it is critical. In this paper, we correspondingly introduce two such definitions: (i) a set $S\subseteq V(G)$ is called a \emph{strong critical vertex-set} of $G$ if $γ(G-S)=γ(G)-|S|$; (ii) a graph $G$ is called \emph{strong $l$-vertex-sets-critical} if $V(G)$ can be partitioned into $l$ strong critical vertex-sets of $G$. Whereafter, we give some properties of strong $l$-vertex-sets-critical graph…
Independent mutual-visibility sets and distance edge-critical graphs
Zhejiang University of Science and Technology（浙江科技学院）；University of Ljubljana（卢布尔雅那大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；University of Maribor（马里博尔大学）
A critical set in an $n \times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\times n$ Latin square and no proper subset of $C$ has this property. The cardinality of the largest critical set in any Latin square of order $n$ is denoted by $\lcs{n}$. We give a lower bound for $\lcs{n}$ by showing that $\lcs{n} \geq n^2(1-\frac{2 + \ln 2}{\ln n})+n(1+\frac {\ln (8 π)} {\ln n})-\frac{\ln 2}{\ln n}.$
In 1998, Khodkar showed that the minimal critical set in the Latin square corresponding to the elementary abelian 2-group of order 16 is of size at most 124. Since the paper was published, improved methods for solving integer programming problems have been developed. Here we give an example of a critical set of size 121 in this Latin square, found through such methods. We also give a new upper bound on the size of critical sets of minimal size for the elementary abelian 2-group of order $2^n$: $4^{n}-3^{n}+4-2^{n}-2^{n-2}$. We speculate about possible lower bounds for this value, given some ot…
A critical set in an $n \times n$ Latin square is a minimal set of entries that uniquely identifies it among all Latin squares of the same size. It is conjectured by Nelder in 1979, and later independently by Mahmoodian, and Bate and van Rees that the size of the smallest critical set is $\lfloor n^2/4\rfloor$. We prove a lower-bound of $n^2/10^4$ for sufficiently large $n$, and thus confirm the quadratic order predicted by the conjecture. We prove a lower-bound of $n^2/10^4$ for sufficiently large $n$, and thus confirm the quadratic order predicted by the conjecture. This improves a recent lo…
Inequalities Involving Core, Corona, and Critical Sets in General Graphs
Instituto de Matemática Aplicada San Luis, Universidad Nacional de San Luis and CONICET（圣路易斯应用数学研究所，圣路易斯国立大学和国家科学研究委员会）；Departamento de Matemática, Universidad Nacional de San Luis（数学系，圣路易斯国立大学）
For $n\leq d$, a family ${\cal F}=\{C_0,C_1,\ldots, C_n\}$ of compact convex sets in $R^d$ is called an $n$-critical family provided any $n$ members of ${\cal F}$ have a non-empty intersection, but $\bigcap_{i=0}^n C_i=\varnothing$. If $n=d$ then a lemma on the intersection of convex sets due to Klee implies that the $d+1$ members of the $d$-critical family enclose a `hollow' in $R^d$, a bounded connected component of $R^d\setminus\bigcup_{i=0}^n C_i.$ Here we prove that the closure of the convex hull of a hollow in $R^d$ is a $d$-simplex.
For a matroid $M$ having $m$ rank-one flats, the density $d(M)$ is $\tfrac{m}{r(M)}$ unless $m = 0$, in which case $d(M)= 0$. A matroid is density-critical if all of its proper minors of non-zero rank have lower density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among simple matroids that cannot be covered by $k$ independent sets is density-critical. It is straightforward to show that $U_{1,k+1}$ is the only minor-minimal loopless matroid with no covering by $k$ independent sets. We prove that there are exactly ten minor-minimal simple obstructions to a matroid being able t…
Given a set $\mathcal{H}$ of graphs, let $f_\mathcal{H}^\star\colon \mathbb{N}_{>0}\to \mathbb{N}_{>0}$ be the optimal $χ$-binding function of the class of $\mathcal{H}$-free graphs, that is, $$f_\mathcal{H}^\star(ω)=\max\{χ(G): G\text{ is } \mathcal{H}\text{-free, } ω(G)=ω\}.$$ In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal $χ$-binding functions for subclasses of $P_5$-free graphs and of $(C_5,C_7,\ldots)$-free graphs. In particular, we prove the following for each $ω\geq 1$: (i) $\ f_{\{P_5,banner\}}^\star(ω)=f_…
Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertices from S are adjacent. By Ind(G) we mean the family of all independent sets of G while core(G) and corona(G) denote the intersection and the union of all maximum independent sets, respectively. The number d(X)= |X|-|N(X)| is the difference of the set of vertices X, and an independent set A is critical if d(A)=max{d(I):I belongs to Ind(G)} (Zhang, 1990). Let ker(G) and diadem(G) be the intersection and union, respectively, of all critical independent sets of G (Levit and Mandrescu, 2012). In this pa…
In this paper we prove: If 0 < d < 1, and p is a sufficiently large prime, then if S is a subset of Z/pZ having the least number of three-term arithmetic progressions among all subsets of Z/pZ having at least dp elements, then S has an arithmetic progression of length at least log^{1/4+o(1)} x.
Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, and mu(G) is the size of a maximum matching. The number id_{c}(G)=max{|I|-|N(I)|:I is an independent set} is called the critical independence difference of G, and A is critical if |A|-|N(A)|=id_{c}(G). We define core(G) as the intersection of all maximum independent sets, and ker(G)as the intersection of all critical independent sets. In this paper we prove that if a graph G is non-quasi-regularizable (i.e., there exists some i…
Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent, alpha(G) is the size of a maximum independent set, and core(G) is the intersection of all maximum independent sets. The number d(X)=|X|-|N(X)| is the difference of the set X, and d_{c}(G)=max{d(I):I is an independent set} is called the critical difference of G. A set X is critical if d(X)=d_{c}(G). For a graph G we define ker(G) as the intersection of all critical independent sets, while diadem(G) is the union of all critical independent sets. For a bipartite graph G=(A,B,E), with bipartition {A,B}, Ore defi…
A set $S\subseteq V(G)$ is independent (or stable) if no two vertices from $S$ are adjacent, and by $\mathrm{Ind}(G)$ we mean the set of all independent sets of $G$. A set $A\in\mathrm{Ind}(G)$ is critical (and we write $A\in CritIndep(G)$) if $\left\vert A\right\vert -\left\vert N(A)\right\vert =\max\{\left\vert I\right\vert -\left\vert N(I)\right\vert :I\in \mathrm{Ind}(G)\}$, where $N(I)$ denotes the neighborhood of $I$. If $S\in\mathrm{Ind}(G)$ and there is a matching from $N(S)$ into $S$, then $S$ is a crown, and we write $S\in Crown(G)$. Let $Ψ(G)$ be the family of all local maximum inde…
We show that an arbitrary infinite graph $G$ can be compactified by its ends plus its critical vertex sets, where a finite set $X$ of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to $X$. We further provide a concrete separation system whose $\aleph_0$-tangles are precisely the ends plus critical vertex sets. Our tangle compactification $\vert G\vert_Γ$ is a quotient of Diestel's (denoted by $\vert G\vert_Θ$), and both use tangles to compactify a graph in much the same way as the ends of a locally finite…
Let G be a simple graph with vertex set V(G). A set S is independent if no two vertices from S are adjacent. The graph G is known to be a Konig-Egervary if alpha(G)+mu(G)= |V(G)|, where alpha(G) denotes the size of a maximum independent set and mu(G) is the cardinality of a maximum matching. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set in G} (Zhang; 1990). Let Omega(G) denote the family of all maximum independent sets. Let us say that a family Gamma of independent sets is a Konig-Egervary collection if |Unio…
Let $G$ be a simple graph with vertex set $V(G)$. A set $S\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. For $X\subseteq V(G)$, the difference of $X$ is $d(X) = |X|-|N(X)|$ and an independent set $A$ is critical if $d(A) = \max \{d(X): X\subseteq V(G) \text{ is an independent set}\}$ (possibly $A=\emptyset$). Let $\text{nucleus}(G)$ and $\text{diadem}(G)$ be the intersection and union, respectively, of all maximum size critical independent sets in $G$. In this paper, we will give two new characterizations of König-Egerváry graphs involving $\text{nucleus}(G)$ and $\te…
A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-λ}$ over a primitive set is maximized by the primes if and only if $λ$ is at least the critical exponent $τ_1 \approx 1.14$. A set is $k$-primitive if no member divides any product of up to $k$ other distinct members.…
Let alpha(G) be the cardinality of a independence set of maximum size in the graph G, while mu(G) is the size of a maximum matching. G is a Konig--Egervary graph if its order equals alpha(G) + mu(G). The set core(G) is the intersection of all maximum independent sets of G (Levit & Mandrescu, 2002). The number def(G)=|V(G)|-2*mu(G) is the deficiency of G (Lovasz & Plummer, 1986). The number d(G)=max{|S|-|N(S)|:S in Ind(G)} is the critical difference of G. An independent set A is critical if |A|-|N(A)|=d(G), where N(S) is the neighborhood of S (Zhang, 1990). In 2009, Larson showed that G is Koni…
Let $G$ be a simple graph with vertex set $V\left( G\right) $. A set $S\subseteq V\left( G\right) $ is independent if no two vertices from $S$ are adjacent, and by $\mathrm{Ind}(G)$ we mean the family of all independent sets of $G$. The number $d\left( X\right) =$ $\left\vert X\right\vert -\left\vert N(X)\right\vert $ is the difference of $X\subseteq V\left( G\right) $, and a set $A\in\mathrm{Ind}(G)$ is critical if $d(A)=\max \{d\left( I\right) :I\in\mathrm{Ind}(G)\}$ (Zhang, 1990). Let us recall the following definitions: $\mathrm{core}\left( G\right) $ = $\bigcap$ {S : S is a maximum indepe…
A set S is independent in a graph G if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. If alpha(G)+mu(G)=|V|, then G=(V,E) is called a Konig-Egervary graph. The number d_{c}(G)=max{|A|-|N(A)|} is called the critical difference of G (Zhang, 1990). By core(G) (corona(G)) we denote the intersection (union, respectively) of all maximum independent sets, while by ker(G) we mean the intersection of all critical independent sets. A connected graph having only one cycle is called …
Fix a prime p and a density 0 < d <= 1. Among all functions f : F_p -> [0,1], what can one say about those which assign minimal weight to three-term arithmetic progressions -- that is, the sum of f(a)f(a+x)f(a+2x) is minimal as we sum over all a and x -- subject to the density constraint that the expected value of f equals d? In the present paper we show three things about them: 1) Such f are nearly indicator functions; 2) They enjoy a certain ``local minimal'' property; and, 3) They are approximately indicator functions for certain sumsets A+B.
Let $G$ be a finite simple graph. For $X \subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\max \, \{d(X):X\subset V(G)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d(X)$ equals the critical difference and ker$(G)$ is the intersection of all critical sets. It is known that ker$(G)$ is an independent (vertex) set of $G$. diadem$(G)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with $d(S) > 0$ if no proper subset of $S$ has positive difference. A graph $G…
Let G=(V,E). A set S is independent if no two vertices from S are adjacent. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set}. Let us recall that ker(G) is the intersection of all critical independent sets, and core(G) is the intersection of all maximum independent sets. Recently, it was established that ker(G) is a subset of core(G) is true for every graph, while the corresponding equality holds for bipartite graphs. In this paper we present various structural properties of ker(G). The main finding claims that …
Let $G$ be a multiplicative group, let $A,B \subseteq G$ be finite and nonempty, and define the product set $AB = {ab \mid $a \in A$ and $b \in B$}$. Two fundamental problems in combinatorial number theory are to find lower bounds on $|AB|$, and then to determine structural properties of $A$ and $B$ under the assumption that $|AB|$ is small. We focus on the extreme case when $|AB| < |A| + |B|$, and call any such pair $(A,B)$ \emph{critical}. In the case when $|G|$ is prime, the Cauchy-Davenport Theorem asserts that $|AB| \ge \min {|G|, |A| + |B| - 1}$, and Vosper refined this result by classif…
The independence number $α(G)$ is the cardinality of a maximum independent set, while $μ(G)$ is the size of a maximum matching in $G$. If $α(G)+μ(G)$ equals the order of $G$, then $G$ is called a Konig-Egervary graph. The number $d\left( G\right) =\max\{\left\vert A\right\vert -\left\vert N\left( A\right) \right\vert :A\subseteq V\}$ is called the critical difference of $G$ (where $N\left( A\right) =\left\{ v:v\in V,N\left( v\right) \cap A\neq\emptyset\right\} $). It is known that $α(G)-μ(G)\leq d\left( G\right) $ holds for every graph. A graph $G$ is unicyclic if it has a unique cycle and alm…
We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $α_c(Δ)$ and provide (i) for $α< α_c(Δ)$ randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most $αn$ in $n$-vertex graphs of maximum degree $Δ$; and (ii) a proof that unless NP=RP, no such algorithms exist for $α>α_c(Δ)$. The critical density is the occupancy fraction of the hard core model on the complete graph $K_{Δ+1}$ at the uniqueness threshold on the…
The problem of leaders location plays an important role in the controllability of undirected graphs.The concept of minimal perfect critical vertex set is introduced by drawing support from the eigenvector of Laplace matrix. Using the notion of minimal perfect critical vertex set, the problem of finding the minimum number of controllable leader vertices is transformed into the problem of finding all minimal perfect critical vertex sets. Some necessary and sufficient conditions for special minimal perfect critical vertex sets are provided, such as minimal perfect critical 2 vertex set, and minim…
