A triangle in a hypergraph $\mathcal{H}$ is a set of three distinct edges $e, f, g\in\mathcal{H}$ and three distinct vertices $u, v, w\in V(\mathcal{H})$ such that $\{u, v\}\subseteq e$, $\{v, w\}\subseteq f$, $\{w, u\}\subseteq g$ and $\{u, v, w\}\cap e\cap f\cap g=\emptyset$. Johansson proved in 1996 that $χ(G)=\mathcal{O}(Δ/\logΔ)$ for any triangle-free graph $G$ with maximum degree $Δ$. Cooper and Mubayi later generalized the Johansson's theorem to all rank $3$ hypergraphs. In this paper we provide a common generalization of both these results for all hypergraphs, showing that if $\mathcal{H}$ is a rank $k$, triangle-free hypergraph, then the list chromatic number \[ χ_{\ell}(\mathcal{H})\leq \mathcal{O}\left(\max_{2\leq \ell \leq k} \left\{\left( \frac{Δ_{\ell}}{\log Δ_{\ell}} \right)^{\frac{1}{\ell-1}} \right\}\right), \] where $Δ_{\ell}$ is the maximum $\ell$-degree of $\mathcal{H}$. The result is sharp apart from the constant. Moreover, our result implies, generalizes and improves several earlier results on the chromatic number and also independence number of hypergraphs, while its proof is based on a different approach than prior works in hypergraphs (and therefore provides alternative proofs to them). In particular, as an application, we establish a bound on chromatic number of sparse hypergraphs in which each vertex is contained in few triangles, and thus extend results of Alon, Krivelevich and Sudakov, and Cooper and Mubayi from hypergraphs of rank 2 and 3, respectively, to all hypergraphs.
We consider the problem of $k$-colouring a random $r$-uniform hypergraph with $n$ vertices and $cn$ edges, where $k$, $r$, $c$ remain constant as $n$ tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case $r=2$, must have one of two easily computable values as $n$ tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs.
A graph with chromatic number $k$ is called $k$-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all triangle-free 5-chromatic graphs up to 24 vertices. This implies that Reed's conjecture holds for triangle-free graphs up to at least this order. We also establish that the smallest regular triangle-free 5-chromatic graphs have 24 vertices. Finally, we show that the smallest 5-chromatic graphs of girth at least 5 have at least 29 vertices and that the smallest 4-chro…
We prove that every subcubic triangle-free graph has fractional chromatic number at most 14/5, thus confirming a conjecture of Heckman and Thomas [A new proof of the independence ratio of triangle-free cubic graphs. Discrete Math. 233 (2001), 233--237].
We give an upper bound on the list chromatic number of a 2-colorable hypergraph which generalizes the bound of Schauz on $k$-partite $k$-uniform hypergraphs. It makes sense for sparse hypergraphs: in particular we show that a $k$-uniform $k$-regular hypergraph has the list chromatic number 2 for $k \geq 4$. Also we obtain both lower and upper bound on the list chromatic number of a complete $s$-uniform 2-colorable hypergraph in the vein of Erd{\H o}s--Rubin--Taylor theorem.
In this paper, we prove that $(-\infty, 0)$ is a zero-free interval for chromatic polynomials of a family ${\cal L}_0$ of hypergraphs and $(0, 1)$ is a zero-free interval for chromatic polynomials of a subfamily ${\cal L}_0'$ of ${\cal L}_0$ of hypergraphs. These results extend known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.
In 1964 Erdős proved, by randomized construction, that the minimum number of edges in a $k$-graph that is not two colorable is $O(k^2\; 2^k)$. To this day, it is not known whether there exist such $k$-graphs with smaller number of edges. Known deterministic constructions use much larger number of edges. The most recent one by Gebauer requires $2^{k+Θ(k^{2/3})}$ edges. Applying derandomization technique we reduce that number to $2^{k+\widetildeΘ(k^{1/2})}$.
We estimate the likely values of the chromatic and independence numbers of the random $r$-uniform $d$-regular hypergraph on $n$ vertices for fixed $r$, large fixed $d$, and $n \rightarrow \infty$.
In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph $G=(V,E)$, the number of totally cyclic orientations of $G$ is equal to the value of $|P(H,-1)|$, where $P(H,λ)$ is the chromatic polynomial of a hypergraph $H$ which is constructed from $G$. Finally we show that the multiplicity of root "$0$" of $P(H,λ)$ may be at least $2$ for…
Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least $c \binom{|V(H)|}{r-1}$ has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erdős-Simonovits. One interesting question, first proposed by Łuczak-Thomassé and then solved by Allen-Böttcher-Griffiths-Kohayakawa-Morris, is the characterization of graphs having zero chromatic threshol…
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 number and the clique number of a comparability 3-hypergraph, this is in contrast with the fact that a comparability simple graph is a perfect graph.
It was conjectured by Ohba and confirmed recently by Noel et al. that, for any graph $G$, if $|V(G)|\le 2χ(G)+1$ then $χ_l(G)=χ(G)$. This indicates that the graphs with high chromatic number are chromatic-choosable. We show that this is also the case for uniform hypergraphs and further propose a generalized version of Ohba's conjecture: for any $r$-uniform hypergraph $H$ with $r\geq 2$, if $|V(H)|\le rχ(H)+r-1$ then $χ_l(H)=χ(H)$. We show that the condition of the proposed conjecture is sharp by giving two classes of $r$-uniform hypergraphs $H$ with $|V(H)|= rχ(H)+r$ and $χ_l(H)>χ(H)$. To supp…
We establish a set of recursion relations for the coefficients in the chromatic polynomial of a graph or a hypergraph. As an application we provide a generalization of Whitney's broken cycle theorem for hypergraphs, as well as deriving an explicit formula for the linear coefficient of the chromatic polynomial of the $r$-complete hypergraph in terms of roots of the Taylor polynomials for the exponential function.
We characterize the r-graph with maximal p-spectral radius among the k-partite r-graphs of order n, and the 3-graph with maximal p-spectral radius among the k-chromatic 3-graphs of order n.
Let $G = (V,E)$ be a finite, simple, connected graph with chromatic polynomial $P_G(q)$. Sokal \cite{sokal} proved that the roots of the chromatic polynomial of $G$ are bounded in absolute value by $KD$ where, $D$ is the maximum degree of the graph and $7< K < 8$ is a constant. In this paper we generalize this result to uniform hypergraphs. To prove our results we will use the theory of the bounded exponential type graph polynomials.
The problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The \emph{$(p,q)$-extremal problem} consists in finding the maximum number of edges on a $k$-uniform hypergraph $\mathcal{H}$ with $n$ vertices such that among any $p$ edges some $q$ of them have no empty intersection. In this paper we have found a link between the fractional chromatic number of Kneser hypergraphs and the $(p,q)$-extremal problem and also solve the $(…
On the largest chromatic number of $F$-free hypergraphs
Tsinghua University（清华大学）；Northwestern Polytechnical University（西北工业大学）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least $c \binom{|V(H)|}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r=2), and in this paper we begin its systematic study for hypergraphs. Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we al…
In 1973 P. Erdős and L. Lovász noticed that any hypergraph whose edges are pairwise intersecting has chromatic number 2 or 3. In the first case, such hypergraph may have any number of edges. However, Erdős and Lovász proved that in the second case, the number of edges is bounded from above. For example, if a hypergraph is $ n $-uniform, has pairwise intersecting edges, and has chromatic number 3, then the number of its edges does not exceed $ n^n $. Recently D.D. Cherkashin improved this bound (see \cite{Ch}). In this paper, we further improve it in the case when the number of vertices of an $…
We consider the following game, played on a $k$-uniform hypergraph $H$. There are $q$ colors available and two players take it in turns to color vertices. A partial coloring is proper if no edge is mono-chromatic. One player, A, wishes to color all the vertices and the other player, B, wishes to prevent this. The {\em game chromatic number} $χ_g(H)$ is the minimum number of colors for which A has a winning strategy. We consider this in the context of a random $k$-uniform hypergraph and prove upper and lower bounds that hold w.h.p.
Recently, Kupavskii~[{\it On random subgraphs of {K}neser and {S}chrijver graphs. J. Combin. Theory Ser. A, {\rm 2016}.}] investigated the chromatic number of random Kneser graphs $\KG_{n,k}(ρ)$ and proved that, in many cases, the chromatic numbers of the random Kneser graph $\KG_{n,k}(ρ)$ and the Kneser graph $\KG_{n,k}$ are almost surely closed. He also marked the studying of the chromatic number of random Kneser hypergraphs $\KG^r_{n,k}(ρ)$ as a very interesting problem. With the help of $\Z_p$-Tucker lemma, a combinatorial generalization of the Borsuk-Ulam theorem, we generalize Kupavskii'…
Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. It is known that for a fixed $n$ one has \[ c_n r^n < m(n,r) < C_n r^n. \] We prove that for any fixed $n$ the sequence $a_r := m(n,r)/r^n$ has a limit, which was conjectured by Alon. We also prove the list colorings analogue of this statement.
Heckman and Thomas conjectured that the fractional chromatic number of any triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami and Zhu and of Lu and Peng, we prove that the fractional chromatic number of any triangle-free subcubic graph is at most 32/11 (which is roughly 2.909).
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge coloring of $G$. It was conjectured that $\chiup'_{a}(G)\leq Δ(G) + 2$ for any simple graph $G$ with maximum degree $Δ(G)$. A graph is {\em $1$-planar} if it can be drawn on the plane such that every edge is crossed by at most one other edge. In this paper, we prove that every triangle-free $1$-planar graph $G$ has an acyclic edge coloring with $Δ(G) + 16$ color…
The family $D(k,m)$ of graphs having an orientation such that for every vertex $v \in V(G)$ either (outdegree) $°^+(v) \le k$ or (indegree) $°^-(v) \le m$ have been investigated recently in several papers because of the role $D(k,m)$ plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family $D(k,m)$ have been obtained via the notion of $d$-degeneracy of graphs. In this paper we consider a far reaching generalization of the family $D(k,m)$, in a complementary form, int…
The generalized Kneser hypergraph $KG^{r}(n,k,s)$ is the hypergraph whose vertices are all the $k$-subsets of $\{1,\ldots ,n\}$, and edges are $r$-tuples of distinct vertices such that any pair of them has at most $s$ elements in their intersection. In this note, we show that for each non-negative integers $k, n, r, s$ satisfying $n \geq r(k-1)+1$, $k > s\geq 0$, and $r\geq 2$, we have $$χ({KG}^{r}(n,k,s))\geq\left\lceil\frac{n-r(k-s-1)}{r-1}\right\rceil,$$ which improves the previously known result by Alon--Frankl--Lovász.
The main purpose of this note is to give a counterexample to the following conjecture, raised by Florian Frick [\textit{Int. Math. Res. Not. IMRN 2020 (13), 4037-4061 (2020)}]. Conjecture. Let $r\geq 3$ and let $\mathcal{F}$ be a set system. Then $$χ\left(\textrm{KG}^r\left(\mathcal{F}_{r-stab}\right)\right)\geq\left\lceil\frac{cd_r\left(\mathcal{F}\right)}{r-1}\right\rceil.$$
In this paper, in view of $Z_p$-Tucker lemma, we introduce a lower bound for chromatic number of Kneser hypergraphs which improves Dol'nikov-K{ř}{\'ı}{ž} bound. Next, we introduce multiple Kneser hypergraphs and we specify the chromatic number of some multiple Kneser hypergraphs. For a vector of positive integers $\vec{s}=(s_1,s_2,\ldots,s_m)$ and a partition $π=(P_1,P_2,\ldots,P_m)$ of $\{1,2,\ldots,n\}$, the multiple Kneser hypergraph ${\rm KG}^r(π; \vec{s};k)$ is a hypergraph with the vertex set $$V=\left\{A:\ A\subseteq P_1\cup P_2\cup\cdots \cup P_m,\ |A|=k, \forall 1\leq i\leq m;\ |A\cap…
Kneser's 1955 conjecture -- proven by Lovász in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser's conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász' result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser's conjecture. Here we prove a common generalization of a…
For any $r$-uniform hypergraph $\mathcal{H}$ with $m$ ($\geq 2$) edges, let $P(\mathcal{H},k)$ and $P_l(\mathcal{H},k)$ be the chromatic polynomial and the list-color function of $\mathcal{H}$ respectively, and let $ρ(\mathcal{H})$ denote the minimum value of $|e\setminus e'|$ among all pairs of distinct edges $e,e'$ in $\mathcal{H}$. We will show that if $r\ge3$, $ρ(\mathcal{H})\ge 2$ and $m\ge \frac{ρ(\mathcal{H})^3}2+1$, then $P_l(\mathcal{H},k)=P(\mathcal{H},k)$ holds for all integers $k\geq \frac{2.4(m-1)}{ρ(\mathcal{H})\log(m-1)}$.
Using a $Z_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a hypergraph).
In the 1970s, Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer $k$, we construct a triangle-free family of line segments in the plane with chromatic number greater than $k$. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number.
More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture were generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier (2016) introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful t…
There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of $\mathbb{Z}_p$-Tucker lemma, Alishahi and Hajiabolhassan [{\it On the chromatic number of general Kneser hypergraphs, Journal of Combinatorial Theory, Series B, 2015}] introduced a lower bound for the chromatic number of Kneser hypergraphs ${\rm KG}^r({\mathcal H})$. Next, Meunier [{\it Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014}] improved their re…
A general Kneser hypergraph ${\rm KG}^r(\mathcal{H})$ is an $r$-uniform hypergraph that somehow encodes the edge intersections of a ground hypergraph $\mathcal{H}$. The colorability defect of $\mathcal{H}$ is a combinatorial parameter providing a lower bound for the chromatic number of ${\rm KG}^r(\mathcal{H})$ which is addressed in a series of works by Dol'nikov [Sibirskii Matematicheskii Zhurnal, 1988}], Kř\'ıž [Transaction of the American Mathematical Society, 1992], and Ziegler~[Inventiones Mathematicae, 2002]. In this paper, we define a new combinatorial parameter, the equitable colorabil…
We study hypergraphs which are uniquely determined by their chromatic, independence and matching polynomials. B. Bollobás, L. Pebody and O. Riordan (2000) conjectured (BPR-conjecture) that almost all graphs are uniquely determined by their chromatic polynomials. We show that for $r$-uniform hypergraphs with $r \geq 3$ this is almost never the case. This disproves the analolgue of the BPR-conjecture for $3$-uniform hypergraphs. For $r =2$ this also holds for the independence polynomial, as shown by J.A. Makowsky and V. Rakita (2017), whereas for the chromatic and matching polynomial this remain…
The Kneser graph $KG_{n,k}$ is the graph whose vertices are the $k$-element subsets of $[n],$ with two vertices adjacent if and only if the corresponding sets are disjoint. A famous result due to Lovász states that the chromatic number of $KG_{n,k}$ is equal to $n-2k+2$. In this paper we discuss the chromatic number of random Kneser graphs and hypergraphs. It was studied in two recent papers, one due to Kupavskii, who proposed the problem and studied the graph case, and the more recent one due to Alishahi and Hajiabolhassan. The authors of the latter paper had extended the result of Kupavskii …
For a hypergraph $H$, define its intersection spectrum $I(H)$ as the set of all intersection sizes $|E\cap F|$ of distinct edges $E,F\in E(H)$. In their seminal paper from 1973 which introduced the local lemma, Erdős and Lovász asked: how large must the intersection spectrum of a $k$-uniform $3$-chromatic intersecting hypergraph be? They showed that such a hypergraph must have at least three intersection sizes, and conjectured that the size of the intersection spectrum tends to infinity with $k$. Despite the problem being reiterated several times over the years by Erdős and other researchers, …
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 quasisymmetric functions $F_S$, but this is not the case for general hypergraphs. We exhibit a class of hypergraphs $H$ --- hypertrees with prime-sized edges --- for which $X_H$ is $F$-positive, and give an explicit combinatorial interpretation for the $F$-coefficients of $X_H$.
For any two non-negative integers h and k, h > k, an L(h, k)-colouring of a graph G is a colouring of vertices such that adjacent vertices admit colours that at least differ by h and vertices that are two distances apart admit colours that at least differ by k. The smallest positive integer δ such that G permits an L(h, k)-colouring with maximum colour δ is known as the L(h, k)-chromatic number (L(h, k)-colouring number) denoted by λ_{h,k}(G). In this paper, we discuss some interesting invariants in hypergraphs. In fact, we study the relation between the spectral gap and L(2, 1)-chromatic numb…
A wheel is a graph consisting of an induced cycle of length at least four and a single additional vertex with at least three neighbours on the cycle. We prove that no Burling graph contains an induced wheel. Burling graphs are triangle-free and have arbitrarily large chromatic number, so this answers a question of Trotignon and disproves a conjecture of Scott and Seymour.
We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjecture…
A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on ver- tices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least positive integer m for which the graph is (m, k)-colourable. All triangle-free graphs on 8 or fewer vertices are (2, 1)-colourable. There are exactly four triangle-free graphs of order 9 which have 1-defective chromatic number 3. We show that these four graphs appear as subgraphs in almost all triangle-free graphs of order 10 with 1-defective chromatic numb…
This paper extends the scenario of the Four Color Theorem in the following way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly) embedded into R^d. We investigate lower and upper bounds on the maximum (weak and strong) chromatic number of hypergraphs in H(d,k). For example, we can prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak chromatic number is Omega(log n/log log n), whereas the weak chromatic number for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.
We prove the s-positivity of the chromatic functions of interval hypergraphs and a conjecture of Taylor~\cite{Taylor15} as a consequence. We also give a new proof of Gasharov's theorem on the s-positivity of the chromatic symmetric functions of $(3+1)$-free posets.
We introduce a new approach and prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $$(1 + o(1)) \frac{1}{3 \sqrt 2} n^{3/2}.$$ We also show a connection to $r$-uniform hypergraphs without (Berge) cycles of length less than six, and estimate their maximum possible size.
A C-coloring of a hypergraph ${\cal H}=(X,{\cal E})$ is a vertex coloring $φ: X\to {\mathbb{N}}$ such that each edge $E\in{\cal E}$ has at least two vertices with a common color. The related parameter $\overlineχ({\cal H})$, called the upper chromatic number of ${\cal H}$, is the maximum number of colors can be used in a C-coloring of ${\cal H}$. A hypertree is a hypergraph which has a host tree $T$ such that each edge $E\in {\cal E}$ induces a connected subgraph in $T$. Notations $n$ and $m$ stand for the number of vertices and edges, respectively, in a generic input hypergraph. We establish …
More than twenty years ago Erdős conjectured~\cite{E1} that a triangle-free graph $G$ of chromatic number $k \geq k_0(\varepsilon)$ contains cycles of at least $k^{2 - \varepsilon}$ different lengths as $k \rightarrow \infty$. In this paper, we prove the stronger fact that every triangle-free graph $G$ of chromatic number $k \geq k_0(\varepsilon)$ contains cycles of $(\frac{1}{64} - \varepsilon)k^2 \log k$ consecutive lengths, and a cycle of length at least $(\tfrac{1}{4} - \varepsilon)k^2 \log k$. As there exist triangle-free graphs of chromatic number $k$ with at most roughly $4k^2 \log k$ v…
We prove that for every $m$ there is a finite point set $\mathcal{P}$ in the plane such that no matter how $\mathcal{P}$ is three-colored, there is always a disk containing exactly $m$ points, all of the same color. This improves a result of Pach, Tardos and Tóth who proved the same for two colors. The main ingredient of the construction is a subconstruction whose points are in convex position. Namely, we show that for every $m$ there is a finite point set $\mathcal{P}$ in the plane in convex position such that no matter how $\mathcal{P}$ is two-colored, there is always a disk containing exact…
Fractional graph isomorphism is the linear relaxation of an integer programming formulation of graph isomorphism. It preserves some invariants of graphs, like degree sequences and equitable partitions, but it does not preserve others like connectivity, clique and independence numbers, chromatic number, vertex and edge cover numbers, matching number, domination and total domination numbers. In this work, we extend the concept of fractional graph isomorphism to hypergraphs, and give an alternative characterization, analogous to one of those that are known for graphs. With this new concept we pro…
