The chromatic number $χ(G)$ of a graph $G$, that is, the smallest number of colors required to color the vertices of $G$ so that no two adjacent vertices are assigned the same color, is a classic and extensively studied parameter. Here we consider the case where $G$ is a random block graph, also known as the stochastic block model. The vertex set is partitioned into $k\in\mathbb{N}$ parts $V_1, \dotsc, V_k$, and for each $1 \le i\le j\le k$, two vertices $u \in V_i, v\in V_j$ are connected by an edge with some probability $p_{ij} \in (0,1)$ independently. Our main result pins down the typical asymptotic value of $χ(G)$ and establishes the distribution of the sizes of the color classes in optimal colorings. We discover that in contrast to the case of a binomial random graph $G(n,p)$, that corresponds to $k=1$ in our model, where the average size of a color class in an (almost) optimal coloring essentially coincides with the independence number, the block model reveals a more diverse picture: the "average" class in an optimal coloring is a convex combination of several types of independent sets that vary in total size as well as in the size of their intersection with each $V_i$, $1\le i \le k$.
In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.
Suppose that two players take turns coloring the vertices of a given graph G with k colors. In each move the current player colors a vertex such that neighboring vertices get different colors. The first player wins this game if and only if at the end, all the vertices are colored. The game chromatic number $χ_g(G)$ is defined as the smallest k for which the first player has a winning strategy. Recently, Bohman, Frieze and Sudakov [Random Structures and Algorithms 2008] analysed the game chromatic number of random graphs and obtained lower and upper bounds of the same order of magnitude. In thi…
A weak $c$-colouring of a balanced incomplete block design (BIBD) is a colouring of the points of the design with $c$ colours in such a way that no block of the design has all of its vertices receive the same colour. A BIBD is said to be weakly $c$-chromatic if $c$ is the smallest number of colours with which the design can be weakly coloured. In this paper we show that for all $c \geq 2$ and $k \geq 3$ with $(c,k) \neq (2,3)$, the obvious necessary conditions for the existence of a $(v,k,λ)$-BIBD are asymptotically sufficient for the existence of a weakly $c$-chromatic $(v,k,λ)$-BIBD.
Let $G$ be a simple graph with maximum degree $Δ(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Δ(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $Δ(G)>n/3$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, Kühn and Osthus is affirmative.
We study a very large family of graphs, the members of which comprise disjoint paths of cliques with extremal cliques identified. This broad characterisation naturally generalises those of various smaller families of graphs having well-known chromatic polynomials. We derive a relatively simple formula for an arbitrary member of the subfamily consisting of those graphs whose constituent clique-paths have at least one trivial extremal clique, and use this formula to show that the set of all non-integer chromatic roots of these graphs is closed under multiplication by natural numbers. A well-know…
The clique chromatic number of a graph is the minimum number of colors required to assign to its vertex set so that no inclusion maximal clique is monochromatic. McDiarmid, Mitsche and Prałat proved that the clique chromatic number of the binomial random graph $G\left(n,\frac{1}{2}\right) $ is at most $\left(\frac{1}{2}+o(1)\right)\log_2n$ with high probability. Alon and Krivelevich showed that it is greater than $\frac{1}{2000}\log_2n$ with high probability and suggested that the right constant in front of the logarithm is $\frac{1}{2}.$ We prove their conjecture and, beyond that, obtain a ti…
We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for $t\leq \frac38p+1$, a small …
Let $G$ be a graph with $n$ vertices, $m$ edges, average degree $δ$, and maximum degree $Δ$. The "oriented chromatic number" of $G$ is the maximum, taken over all orientations of $G$, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which $δ\geq\log n$. We prove that every such graph has oriented chromatic number at least $Ω(\sqrt{n})$. In the case that $δ\geq(2+ε)\log n$, this lower bound is improved to $Ω(\sqrt{m})…
Let $k$ and $\ell$ be positive integers. A cycle with two blocks $c(k,\ell)$ is an oriented cycle which consists of two internally (vertex) disjoint directed paths of lengths at least $k$ and $\ell$, respectively, from a vertex to another one. A problem of Addario-Berry, Havet and Thomassé (2007) asked if, given positive integers $k$ and $\ell$ such that $k+\ell\ge 4$, any strongly connected digraph $D$ containing no $c(k,\ell)$ has chromatic number at most $k+\ell-1$. In this paper, we show that such digraph $D$ has chromatic number at most $O((k+\ell)^2)$, improving the previous upper bound …
The chromatic number $χ(G)$ of a graph $G$ is defined as the minimum number of colours required for a vertex colouring where no two adjacent vertices are coloured the same. The chromatic number of the dense random graph $G \sim G(n,p)$ where $p \in (0,1)$ is constant has been intensively studied since the 1970s, and a landmark result by Bollobás in 1987 first established the asymptotic value of $χ(G)$. Despite several improvements of this result, the exact value of $χ(G)$ remains open. In this paper, new upper and lower bounds for $χ(G)$ are established. These bounds are the first ones that ma…
An equitable colouring of a graph $G$ is a colouring of the vertices of $G$ so that no two adjacent vertices are coloured the same and, additionally, the colour class sizes differ by at most $1$. The equitable chromatic number $χ_=(G)$ is the minimum number of colours required for this. We study the equitable chromatic number of the dense random graph $G(n,m)$, where $m = \left\lfloor p {n \choose 2} \right \rfloor $ and $0<p< 0.86$ is constant. It is a well-known question of Bollobás whether for $p=1/2$ there is a function $f(n) \rightarrow \infty$ so that for any sequence of intervals of len…
Given a group $G$ of automorphisms of a graph $Γ$, the orbital chromatic polynomial $OP_{Γ,G}(x)$ is the polynomial whose value at a positive integer $k$ is the number of orbits of $G$ on proper $k$-colorings of $Γ.$ In \cite{Cameron}, Cameron et. al. explore the roots of orbital chromatic polynomials, and in particular prove that orbital chromatic roots are dense in $\mathbb{R}$, extending Thomassen's famous result (see \cite{Thomassen}) that chromatic roots are dense in $[\frac{32}{27},\infty)$. Cameron et al \cite{Cameron} further conjectured that the real roots of the orbital chromatic pol…
The chromatic threshold $δ_χ(H,p)$ of a graph $H$ with respect to the random graph $G(n,p)$ is the infimum over $d > 0$ such that the following holds with high probability: the family of $H$-free graphs $G \subset G(n,p)$ with minimum degree $δ(G) \ge dpn$ has bounded chromatic number. The study of the parameter $δ_χ(H) := δ_χ(H,1)$ was initiated in 1973 by Erdős and Simonovits, and was recently determined for all graphs $H$. In this paper we show that $δ_χ(H,p) = δ_χ(H)$ for all fixed $p \in (0,1)$, but that typically $δ_χ(H,p) \ne δ_χ(H)$ if $p = o(1)$. We also make significant progress towa…
A path with three blocks $P(k,l,r)$ is an oriented path formed by $k$-forward arcs followed by $l$-backward arcs then $r$-forward arcs. We prove that any $(2k+1)$-chromatic digraph contains a path $P(1,k,1)$. However the existence of $P(1,l,1)$ with $l \geq k$ is established in any $(k+4)$-chromatic digraph. In general, we establish a quadratic bound for paths with three blocks.
The chromatic number of a digraph $D$ is the minimum number of acyclic subgraphs covering the vertex set of $D$. A tournament $H$ is a hero if every $H$-free tournament $T$ has chromatic number bounded by a function of $H$. Inspired by the celebrated Erdős--Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. We extend this framework to dense digraphs: A digraph $H$ is a superhero if every $H$-free digraph $D$ has chromatic number bounded by a function of $H$ and $α(D)$, the independence number of the underlying graph of $D$. We prove here that a digraph is a super…
The clique chromatic number of a graph G=(V,E) is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random graph G=G(n,1/2) is, with high probability, Ω(log n). This settles a problem of McDiarmid, Mitsche and Pralat who proved that it is O(log n) with high probability.
A $(2+1)$-bispindle $B(k_1,k_2;k_3)$ is the union of two $xy$-dipaths of respective lengths $k_1$ and $k_2$, and one $yx$-dipath of length $k_3$, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. conjectured that, for every positive integers $k_1, k_2, k_3$, there is an integer $g(k_1, k_2, k_3)$ such that every strongly connected digraph not containing subdivisions of $B(k_1, k_2; k_3)$ has a chromatic number at most $g(k_1, k_2, k_3)$, and they proved it only for the case where $k_2=1$. For Hamiltonian digraphs, we prove Cohen et al.'s conjecture, namely $g(k_1, k_…
We consider the structure of $H$-free subgraphs of graphs with high minimal degree. We prove that for every $k>m$ there exists an $ε:=ε(k,m)>0$ so that the following holds. For every graph $H$ with chromatic number $k$ from which one can delete an edge and reduce the chromatic number, and for every graph $G$ on $n>n_0(H)$ vertices in which all degrees are at least $(1-ε)n$, any subgraph of $G$ which is $H$-free and contains the maximum number of copies of the complete graph $K_m$ is $(k-1)$-colorable. We also consider several extensions for the case of a general forbidden graph $H$ of a given …
Let $G$ be a simple connected graph of order $n$. A hamiltonian coloring $c$ of a graph $G$ is an assignment of colors (non-negative integers) to the vertices of $G$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n - 1$ for every two distinct vertices $u$ and $v$ of $G$, where $D(u, v)$ denotes the detour distance between $u$ and $v$ in $G$ which is the length of the longest path between $u$ and $v$. The value \emph{hc(c)} of a hamiltonian coloring $c$ is the maximum color assigned to a vertex of $G$. The hamiltonian chromatic number, denoted by $hc(G)$, is min\{$hc(c)$\} taken over all hamilto…
An equitable coloring of a graph $G$ is a proper vertex coloring of $G$ such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Goldberg conjecture for multigraphs. The results obtained in the paper support our con…
For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of edges between parts of a partition of vertex set of $G$ into two parts such that each part has size at least $r$. We show that if $t=(1-o(1))n$ and $n$ is large enough, then for any dense graph $G$ with $n$ vertices, the chromatic number of the tree graph ${\cal T}_{G,t}$ is equal to the $(n-t+1)^{th}$ cut number of $G$. In particular, as a consequence, we prov…
An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any $C$ a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree $2$ and two vertices have in-degree $2$).
The Vapnik-Červonenkis dimension is a complexity measure of set-systems, or hypergraphs. Its application to graphs is usually done by considering the sets of neighborhoods of the vertices (cf. Alon et al. (2006) and Chepoi, Estellon, and Vaxes (2007)), hence providing a set-system. But the graph structure is lost in the process. The aim of this paper is to introduce the notion of paired VC-dimension, a generalization of VC-dimension to set-systems endowed with a graph structure, hence a collection of pairs of subsets. The classical VC-theory is generally used in combinatorics to bound the tran…
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…
I show that the zeros of the chromatic polynomials P_G(q) for the generalized theta graphs Θ^{(s,p)} are, taken together, dense in the whole complex plane with the possible exception of the disc |q-1| < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate corollary is that the chromatic zeros of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for…
If $G$ is a $k$-chromatic graph of order $n$ then it is known that the chromatic polynomial of $G$, $π(G,x)$, is at most $x(x-1)\cdots (x-(k-1))x^{n-k} = (x)_{\downarrow k}x^{n-k}$ for every $x\in \mathbb{N}$. We improve here this bound by showing that \[ π(G,x) \leq (x)_{\downarrow k} (x-1)^{Δ(G)-k+1} x^{n-1-Δ(G)}\] for every $x\in \mathbb{N},$ where $Δ(G)$ is the maximum degree of $G$. Secondly, we show that if $G$ is a connected $k$-chromatic graph of order $n$ where $k\geq 4$ then $π(G,x)$ is at most $(x)_{\downarrow k}(x-1)^{n-k}$ for every real $x\geq n-2+\left( {n \choose 2} -{k \choose…
A hamiltonian coloring c of a graph G of order p is an assignment of colors to the vertices of G such that $D(u,v)+|c(u)-c(v)|\geq p-1$ for every two distinct vertices u and v of G, where D(u,v) denoted the detour distance between u and v. The value hc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number, denoted by hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we present a lower bound for the hamiltonian chromatic number of block graphs and give a sufficient condition to achieve the lower bound. W…
Resolving a conjecture of Bollobás and Erdős, Gyárfás proved that every graph $G$ of chromatic number $k+1\geq 3$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ distinct odd lengths. We strengthen this prominent result by showing that such $G$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ consecutive odd lengths. Along the way, combining extremal and structural tools, we prove a stronger statement that every graph of chromatic number $k+1\geq 7$ contains $k$ cycles of consecutive lengths, except that some block is $K_{k+1}$. As corollaries, this confirms a conjecture of Verstraëte and answers…
In 1995, Stanley introduced the well-known chromatic symmetric function $X_{G}(x_{1},x_{2},\ldots)$ of a graph $G$. It is a sum of monomial symmetric functions such that for each vertex coloring of $G$ there is exactly one of these summands. The question, whether $X_{G}(x_1,x_{2},\ldots)$ distinguishes nonisomorphic trees with the same number of vertices, is still open in general. For special trees it has already been shown. In 2008, Martin, Morin and Wagner proved it for spiders and some caterpillars. We decompose a tree by separating the leafs and their neighbors and do the same to the remai…
It is known that every graph of sufficiently large chromatic number and bounded clique number contains, as an induced subgraph, a subdivision of any fixed forest, and a subdivision of any fixed cycle. Equivalently, forests and triangles are pervasive, where H is pervasive (in some class of graphs) if for all s>0, every graph in the class with bounded clique number and sufficiently large chromatic number contains an induced subdivision of H, with every edge subdivided at least s times. Which other graphs are pervasive? Chalopin, Esperet, Li and Ossona de Mendez proved that every such graph is a…
Motivated by frequency assignment in office blocks, we study the chromatic number of the adjacency graph of $3$-dimensional parallelepiped arrangements. In the case each parallelepiped is within one floor, a direct application of the Four-Colour Theorem yields that the adjacency graph has chromatic number at most $8$. We provide an example of such an arrangement needing exactly $8$ colours. We also discuss bounds on the chromatic number of the adjacency graph of general arrangements of $3$-dimensional parallelepipeds according to geometrical measures of the parallelepipeds (side length, total …
We construct dense, triangle-free, chromatic-critical graphs of chromatic number $k$ for all $k\geq 4$. For $k\geq 6$ our constructions have $> (\frac{1}{4} -\varepsilon)n^2$ edges, which is asymptotically best possible by Turán's theorem. We also demonstrate (nonconstructively) the existence of dense $k$-critical graphs avoiding all odd cycles of length $\leq \ell$ for any $\ell$ and any $k\geq 4$, again with a best possible density of $>(\frac{1}{4} -\varepsilon)n^2$ edges for $k\geq 6$. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the c…
Given a graph $G$, a coloring $c:V(G)\longrightarrow \{1,\ldots,k\}$ such that $c(u)=c(v)=i$ implies that vertices $u$ and $v$ are at distance greater than $i$, is called a packing coloring of $G$. The minimum number of colors in a packing coloring of $G$ is called the packing chromatic number of $G$, and is denoted by $χ_ρ(G)$. In this paper, we propose the study of $χ_ρ$-critical graphs, which are the graphs $G$ such that for any proper subgraph $H$ of $G$, $χ_ρ(H)<χ_ρ(G)$. We characterize $χ_ρ$-critical graphs with diameter 2, and $χ_ρ$-critical block graphs with diameter 3. Furthermore, we…
In this paper, we provide a natural bijection between a special family of block circulant graphs and the graphs of critical pairs of the posets known as generalized crowns. In particular, every graph in this family of block circulant graphs we investigate has a generating block row that follows a symmetric growth pattern of the all ones matrix. The natural bijection provides an upper bound on the chromatic number for this infinite family of graphs.
The celebrated Erdős--Stone--Simonovits theorem characterizes the asymptotic maximum edge density in $\mathcal{F}$-free graphs as $1 - 1/(χ(\mathcal{F})-1) + o(1)$, where $χ(\mathcal{F})$ is the minimum chromatic number of a graph in $\mathcal{F}$. In Examples 25 and 31 of [L. N. Coregliano and A. A. Razborov. Semantic limits of dense combinatorial objects. Uspekhi Mat. Nauk, 75(4(454)):45-152, 2020], it was shown that this result can be extended to the general setting of graphs with extra structure: the maximum asymptotic density of a graph with extra structure without some induced subgraphs …
Problem of finding an optimal upper bound for the chromatic no. of a graph is still open and very hard. Borodin and Kostochka Conjecture is still open and if proved will improve Brook bound on Chromatic no. of a graph. Here we prove Borodin & Kostochka Conjecture for (1) (P4 Union K1)-free (2) P5-free (3) Chair-free graphs and 4) graphs with dense neighbourhoods. Certain known results follow as Corollaries.
The Bandwidth theorem of Böttcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ that ensures $G$ contains every $r$-chromatic graph $H$ on $n$ vertices of bounded degree and of bandwidth $o(n)$, thereby proving a conjecture of Bollobás and Komlós. In this paper we prove a version of the Bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense $n$-vertex graph $G$ with $δ(G) > (1/2+o(1))n$ contains as a subgraph any given (spanning) $H$ with bounded maximum degree and sublinear bandwidth.
A variety of the mutual-visibility coloring problem for graphs
Cochin University of Science and Technology（科钦科技大学）；University of Maribor（马里博尔大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；Universidad de Cádiz（加的斯大学）
We study the two-player game where Maker and Breaker alternately color the edges of a given graph $G$ with $k$ colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, all edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored edge where every color is blocked. The game chromatic index $χ'_g(G)$ denotes the smallest $k$ for which Maker has a winning strategy. The trivial bounds $Δ(G) \le χ_g'(G) \le 2Δ(G)-1$ hold for every graph $G$, where $Δ(G)$ is the maximum degree of $G$. In 2008, Beveridge, Bohman, Frieze, an…
The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1| < 1.…
Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each V_i. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by adding k \lceil n/k \rceil - n isolated vertices is strongly k-colorable. The strong chromatic number of G is the minimum k for which G is strongly k-colorable. In this paper, we study the behavior…
