Suppose that $n \ge 2$, and we wish to plant $k$ different types of trees in the squares of an $n \times n$ square grid. We can have as many of each type as we want. The only rule is that every pair of types must occur in an adjacent pair of squares somewhere in the grid. The question is: given $n$, what is the largest that $k$ can be? Denote this number by $Γ(n)$, and call this the *complete coloring number* of the $n \times n$ grid. A little thought shows that $Γ(n) \le 2n-1$. The main question we are interested in is whether $Γ(n) = 2n-1$ for every $n \ge 2$.
We study the infinite graph of $n$-dimensional rectangular grid that doesn't appear distance regular and the distance regular colorings of this graph, which are defined as the distance colorings with respect to completely regular codes. It is proved that the elements of the parameter matrix of an arbitrary distance regular coloring form two monotonic sequences. It is shown that every irreducible distance regular coloring of the $n$-dimensional rectangular grid has at most $2n+1$ colors.
A two-dimensional \emph{grid} is a set $\Gnm = [n]\times[m]$. A grid $\Gnm$ is \emph{$c$-colorable} if there is a function $χ_{n,m}: \Gnm \to [c]$ such that there are no rectangles with all four corners the same color. We address the following question: for which values of $n$ and $m$ is $\Gnm$ $c$-colorable? This problem can be viewed as a bipartite Ramsey problem and is related to a the Gallai-Witt theorem (also called the multidimensioanl Van Der Waerden's Theorem). We determine (1) \emph{exactly} which grids are 2-colorable, (2) \emph{exactly} which grids are 3-colorable, and (3) \emph{exa…
A star edge-coloring of a graph $G$ is a proper edge-coloring without bichromatic paths or cycles of length four. The smallest integer $k$ such that $G$ admits a star edge-coloring with $k$ colors is the star chromatic index of $G$. In the seminal paper on the topic, Dvořák, Mohar, and Šámal asked if the star chromatic index of complete graphs is linear in the number of vertices and gave an almost linear upper bound. Their question remains open, and consequently, to better understand the behavior of the star chromatic index, this parameter has been studied for a number of other classes of grap…
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 the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of the vertices in which no two vertices at distance $2$ receive the same color, an injective coloring is a coloring of the vertices in which no two vertices with a common neighbor receive the same color, and an exact square coloring is a coloring of the vertices in which no two vertices at distance exactly $2$ receive the same color. We prove that planar graph…
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we first obtain some upper bounds for the star chromatic index of the Cartesian product of two graphs. We then determine the exact value of the star chromatic index of $2$-dimensional grids. We also obtain some upper bounds on the star chromatic index of the Cartesian product of a path with a cyc…
An \textit{$r$-dynamic $k$-coloring} of a graph $G$ is a proper $k$-coloring of $G$ such that every vertex in $V(G)$ has neighbors in at least $\min\{d(v),r\}$ different color classes. The \textit{$r$-dynamic chromatic number} of a graph $G$, written $χ_r(G)$, is the least $k$ such that $G$ has such a coloring. Proving a conjecture of Jahanbekam, Kim, O, and West, we show that the $m$-by-$n$ grid has no $3$-dynamic $4$-coloring when $mn\equiv2\mod 4$. This completes the determination of the $r$-dynamic chromatic number of the $m$-by-$n$ grid for all $r,m,n$.
We say that a vertex-coloring of a graph is a proper k-distance domatic coloring if for each color, every vertex is within distance k from a vertex receiving that color. The maximum number of colors for which such a coloring exists is called the k-distance domatic number of the graph. The problem of determining the k-distance domatic number is motivated by questions about multi-agent networks including arrangements of sensors and robotics. Here, we find the exact k-distance domatic numbers for all grid graphs formed from the Cartesian product of two sufficiently long paths.
Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32Δ+1$ colors are sufficient to square color every planar graph of maximum degree $Δ$. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that $2Δ+7$ colors are sufficient, which improves the best known bounds when $6\leqslant Δ\leqslant 31$.
We consider the problem of $2$-coloring geometric hypergraphs. Specifically, we show that there is a constant $m$ such that any finite set of points in the plane $\mathcal{S} \subset {\mathbb R}^2$ can be $2$-colored such that every axis-parallel square that contains at least $m$ points from $\mathcal{S}$ contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a $2$-coloring. By affine transformations this result immediately applies also when considering $2$-coloring points with respect to homothets of a fixed parallelogram…
The face-centered cubic grid is a three dimensional 12-regular infinite grid. This graph represents an optimal way to pack spheres in the three-dimensional space. In this grid, the vertices represent the spheres and the edges represent the contact between spheres. We give lower and upper bounds on the chromatic number of the d th power of the face-centered cubic grid. In particular, in the case d = 2 we prove that the chromatic number of this grid is 13. We also determine sharper bounds for d = 3 and for subgraphs of of the face-centered cubic grid.
We study vertex colorings of the square $G^2$ of an outerplanar graph $G$. We find the optimal bound of the inductiveness, chromatic number and the clique number of $G^2$ as a function of the maximum degree $Δ$ of $G$ for all $Δ\in \nats$. As a bonus, we obtain the optimal bound of the choosability (or the list-chromatic number) of $G^2$ when $Δ\geq 7$. In the case of chordal outerplanar graphs, we classify exactly which graphs have parameters exceeding the absolute minimum.
The 600 cell S has exactly 10 5-colorings. From these colorings we can construct the space of colorings $B(S)$. This complex has 1344 colorings, and is isomorphic to the space of 5 by 5 Latin Squares. These simplices split into 4 copies of a quotient of S by an involution, and two copies of a space made up of even Latin Squares.
A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, such that two adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance between to vertices and so a more general coloring concept can be defined: The strong coloring of a graph. So a k-strong coloring is a coloring where two vertices may not have the same color if their distance to each other is at most k. The 2-strong coloring of the line graph is known as the strong edge coloring. Coloring the kth power G^k of a graph G is the same as finding a k-strong …
Let $G$ be a planar graph without 4-cycles and 5-cycles and with maximum degree $Δ\ge 32$. We prove that $χ_{\ell}(G^2)\le Δ+3$. For arbitrarily large maximum degree $Δ$, there exist planar graphs $G_Δ$ of girth 6 with $χ(G_Δ^2)=Δ+2$. Thus, our bound is within 1 of being optimal. Further, our bound comes from coloring greedily in a good order, so the bound immediately extends to online list-coloring. In addition, we prove bounds for $L(p,q)$-labeling. Specifically, $λ_{2,1}(G)\le Δ+8$ and, more generally, $λ_{p,q}(G)\le (2q-1)Δ+6p-2q-2$, for positive integers $p$ and $q$ with $p\ge q$. Again, …
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.
A proper vertex coloring of a graph $G$ is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored $P_4$ in $G$. In this paper, we show that the Cartesian product of any two cycles, except $C_3 \square C_3$ and $C_3 \square C_5$, has a $5$-star coloring.
The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $χ(H)$ and $χ_l(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $χ_l (H) = χ(H)$. It is an interesting problem to find graphs that are chromatic-choosable. Kostochka and Woodall \cite{KW2001} conjectured that $χ_l(G^2) = χ(G^2)$ for every graph $G$, which is called List Square Coloring Conjecture. In this paper, we give infinitely man…
For a graph $G(V,E)$ and $l \in \mathbb{N}$, an $l$ distance coloring is a coloring $f: V \to \{1, 2, \cdots, n\}$ of $V$ such that $\forall u,\;v \in V,\; u\neq v,\; f(u)\neq f(v)$ when $d(u,v) \leq l$. Here $d(u,v)$ is the distance between $u$ and $v$ and is equal to the minimum number of edges that connect $u$ and $v$ in $G$. The span of $l$ distance coloring of $G$, $λ^{l}(G)$, is the minimum $n$ among all $l$ distance coloring of $G$. A class of channel assignment problem in cellular network can be formulated as a distance graph coloring problem in regular grid graphs. The cellular networ…
The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $χ(H)$ and $χ_{\ell}(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $χ_{\ell} (H) = χ(H)$. It is an interesting problem to find graphs that are chromatic-choosable. Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable …
A proper vertex $k$-coloring of a graph $G=(V,E)$ is an assignment $c:V\to \{1,2,\ldots,k\}$ of colors to the vertices of the graph such that no two adjacent vertices are associated with the same color. The square $G^2$ of a graph $G$ is the graph defined by $V(G)=V(G^2)$ and $uv \in E(G^2)$ if and only if the distance between $u$ and $v$ is at most two. We denote by $χ(G^2)$ the chromatic number of $G^2$, which is the least integer $k$ such that a $k$-coloring of $G^2$ exists. By definition, at least $Δ(G)+1$ colors are needed for this goal, where $Δ(G)$ denotes the maximum degree of the grap…
For a graph $G$, by $χ_2(G)$ we denote the minimum integer $k$, such that there is a $k$-coloring of the vertices of $G$ in which vertices at distance at most 2 receive distinct colors. Equivalently, $χ_2(G)$ is the chromatic number of the square of $G$. In 1977 Wegner conjectured that if $G$ is planar and has maximum degree $Δ$, then $χ_2(G) \leq 7$ if $Δ\leq 3$, $χ_2(G) \leq Δ+5$ if $4 \leq Δ\leq 7$, and $\lfloor 3Δ/2 \rfloor +1$ if $Δ\geq 8$. Despite extensive work, the known upper bounds are quite far from the conjectured ones, especially for small values of $Δ$. In this work we show that …
A classical question in combinatorics is the following:\ given a partial Latin square $P$, when can we complete $P$ to a Latin square $L$? In this paper, we investigate the class of \textbf{$ε$-dense partial Latin squares}:\ partial Latin squares in which each symbol, row, and column contains no more than $εn$-many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and Häggkvist conjectured that all $\frac{1}{4}$-dense partial Latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a…
We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted $χ^{\sharp 2}(G)$. This notion is related to other types of distance-based colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide. We prove tight bounds on special subclasses of planar graphs: subcubic bipartite pla…
The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of an $n$ elements set, with two vertices adjacent if they are disjoint. The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Determining the chromatic number of the square of the Kneser graph $K(n, k)$ is an interesting graph coloring problem, and is also related with intersecting family problem. The square of $K(2k, k)$ is a perfect matching and the square of $K(n, k)$ is the complete grap…
DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. Motivated by results related to list coloring Cartesian products of graphs, we initiate the study of the DP-chromatic number, $χ_{DP}$, of the same. We show that $χ_{DP}(G \square H) \leq \text{min}\{χ_{DP}(G) + \text{col}(H), χ_{DP}(H) + \text{col}(G) \} - 1$ where $\text{col}(H)$ is the coloring number of the graph $H$. We focus on building tools for lower bound arguments for $χ_{DP}(G \square H)$ and use them to show the sharpness of the bound above and its various…
A connected simple graph is said dual-hamiltonian if its vertex set has a $2$-coloring such that each color class induces a tree. We call such a coloring a hamiltonian coloring. We prove that if $G$ is a graph with a certain type of hamiltonian coloring and $T$ is a tree, then $G\times T$ is also dual-hamiltonian having the same certain type of hamiltonian coloring. This result is used to constructed a class of dual-hamiltonian graphs, which includes the hypercubes and other multidimensional grids.
For a graph $G$, let $G^2$ be the graph with the same vertex set as $G$ and $xy \in E(G^2)$ when $x \neq y$ and $d_G(x,y) \leq 2$. Bonamy, Lévêque, and Pinlou conjectured that if $mad (G) < 4 - \frac{2}{c+1}$ and $Δ(G)$ is large, then $χ_\ell(G^2) \leq Δ(G) + c$. We prove that if $c \geq 3$, $mad (G) < 4 - \frac{4}{c+1}$, and $Δ(G)$ is large, then $χ_\ell(G^2) \leq Δ(G) + c$. Dvořák, Krá\soft{l}, Nejedlý, and Škrekovski conjectured that $χ(G^2) \leq Δ(G) +2$ when $Δ(G)$ is large and $G$ is planar with girth at least $5$; our result implies $χ(G^2) \leq Δ(G) +6$.
We consider the problem of extending partial edge colorings of cartesian products of graphs. More specifically, we suggest the following Evans-type conjecture: If $G$ is a graph where every precoloring of at most $k$ precolored edges can be extended to a proper $χ'(G)$-edge coloring, then every precoloring of at most $k+1$ edges of $G \square K_2$ is extendable to a proper $(χ'(G) +1)$-edge coloring of $G \square K_2$. In this paper we verify that this conjecture holds for trees, complete and complete bipartite graphs, as well as for graphs with small maximum degree. We also prove versions of …
We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if $G$ is such a graph, then $G^2$ is $(Δ(G)+72)$-degenerate. This implies an upper bound of $Δ(G)+73$ on the chromatic number of $G^2$ as well as on several variants of the chromatic number such as the list-chromatic number, paint number, Alon--Tarsi number, and correspondence chromatic number. We also show that if $Δ(G)$ is sufficiently large, then the upper bounds on each of these parameters of $G^2$ can all be lowered to $Δ(G)+2$ (which is best possible). To complement these results, we show…
The discharging method is a powerful proof technique, especially for graph coloring problems. Its major downside is that it often requires lengthy case analyses, which are sometimes given to a computer for verification. However, it is much less common to use a computer to actively look for a discharging proof. In this paper, we use a Linear Programming approach to automatically look for a discharging proof. While our system is not entirely autonomous, we manage to make some progress towards Wegner's conjecture for distance-$2$ coloring of planar graphs, by showing that $12$ colors are sufficie…
Two-colorings of finite grids: variations on a theorem of Tibor Gallai
University of Bucharest, Faculty of Mathematics and Informatics（布加勒斯特大学数学与信息学院）；Research Center for Logic, Optimization and Security (LOS), Faculty of Mathematics and Computer Science, University of Bucharest（布加勒斯特大学数学与计算机科学学院逻辑、优化与安全研究中心）；Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛数学研究所）
We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree $Δ$ are list $(Δ+1)$-colorable, while the squares of some planar graphs of girth six and arbitrarily large maximum degree are not. By Euler's Formula, planar graphs of girth at least $6$ are of maximum average degree less …
A word $\bar{w} = \bar{u}\bar{u}$ is a $long$ $square$ if $\bar{u}$ is of length at least 3; a word $\bar{w}$ is $long$-$square$-$free$ if $\bar{w}$ contains no sub-word that is a long square. We can use words to generate graph colorings; a graph coloring is called $long$-$repetition$-$free$ if the word formed by the coloring of each path in the graph is long-square-free. Our results show that every rooted tree of radius less than or equal to seven is long-repetition-free two-colorable. We also prove there exists a class of trees which are not long-repetition-free two-colorable.
We show that the threshold for the binomial random $3$-partite, $3$-uniform hypergraph $G^{3}((n,n,n),p)$ to contain a Latin square is $Θ(\log{n}/n)$. We also prove analogous results for Steiner triple systems and proper list edge-colorings of the complete (bipartite) graph with random lists. Our results answer several related questions of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus.
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…
Let $χ_{\bar{k}}(n)$ be the number of colors required to color the $n$-dimensional hypercube such that no two vertices with the same color are at a distance at most $k$. In other words, $χ_{\bar{k}}(n)$ is the minimum number of binary codes with minimum distance at least $k+1$ required to partition the $n$-dimensional Hamming space. By giving an explicit coloring, it is shown that $χ_{\bar{2}}(8)=13$.
An edge-coloring of a graph $G$ with colors $1,...,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if $G$ has an interval $t$-coloring for some positive integer $t$. Let $\mathfrak{N}$ be the set of all interval colorable graphs. For a graph $G\in \mathfrak{N}$, the least and the greatest values of $t$ for which $G$ has an interval $t$-coloring are denoted by $w(G)$ and $W(G)$, respectively. In this paper we first show that if $G$ is an $r$-regular grap…
A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two $k$-colorings are $k$-equivalent if we can transform one into the other using Kempe swaps. The triangulated toroidal grid, $T[m\times n]$, is formed from (a toroidal embedding of) the Cartesian product of $C_m$ and $C_n$ by adding parallel diagonals inside all 4-faces. Mohar and Salas showed that not all 4-colorings of $T[m\times n]$ are 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of $T[m\times n]$ are 6-equivalent. They asked whether the…
