We study perfect $2$-coloring of the Johnson graphs $J(n,3)$ associated with the third largest eigenvalue and symmetric quotient matrix, which exists only when $n \in \{6, 10\}$. We survey the known constructions in the case $n=6$, give a new construction for the two known perfect $2$-colorings in the case $n=10$, and prove that these are the only possible ones.
Given a perfect coloring of a graph, we prove that the $L_1$ distance between two rows of the adjacency matrix of the graph is not less than the $L_1$ distance between the corresponding rows of the parameter matrix of the coloring. With the help of an algebraic approach, we deduce corollaries of this result for perfect $2$-colorings, perfect colorings in distance-$l$ graphs and in distance-regular graphs. We also provide examples when the obtained property reject several putative parameter matrices of perfect colorings in infinite graphs.
In this paper we enumerate the parameter matrices of all perfect 2-colorings of the generalized Petersen graphs GP(n;3), where n>=7. We also give some basic results for GP(n; k).
In this paper we obtain the necessary condition for the existence of perfect $k$-colorings (equitable $k$-partitions) in Hamming graphs $H(n,q)$, where $q=2,3,4$ and Doob graphs $D(m,n)$. As an application, we prove the non-existence of extended perfect codes in $H(n,q)$, where $q=3,4$, $n>q+2$, and in $D(m,n)$, where $2m+n>6$.
We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we observe known constructions of perfect colorings and propose some new ones giving new parameters. At last, we deduce which parameters of colorings are covered by these constructions and give tables of admissible parameters of $2$-colorings in Hamming graphs $H(n,q)$ for small $n$ and $q$. Using the connection with perfect colorings, we construct an orthogon…
Under study are the eigenfunctions and perfect colorings of the graph of $n$-dimensional $q$-ary Hamming space. We obtain the interdependence of local distributions of an eigenfunction in two orthogonal faces. We prove also an analogous result for perfect colorings.
A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours of any vertex depends only on the color of the vertex. We consider perfect colorings of Cayley graphs of the additive group of integers with generating set $\{1,-1,3,-3,5,-5,\dots, 2n-1,1-2n\}$ for a positive integer $n$. We enumerate perfect $2$-colorings of the graphs under consideration and state the conjecture generalizing the main result to an arbitrary…
A vertex 2-coloring of a graph is said to be perfect with parameters $(a_{ij})_{i,j=1}^k$ if for every $i,j\in\{1,...,k\}$ every vertex of color $i$ is adjacent with exactly $a_{ij}$ vertices of color $j$. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube $\{0,1\}^{24}$ with parameters $((20+c,256-c)(c,276-c))$ (i.e., with eigenvalue 20). We prove that such colorings exist for all $c$ from 1 to 128 except 1, 2, 4, 5, 7, 10, 13 and do not exist for $c=1, 2, 4, 5, 7$. Keywords: perfect coloring, equitable partition, hypercube, halved n-cube
A coloring of vertices of a given graph is called perfect if the color structure of each ball of radius $1$ in the graph depends only on the color of the ball center. Let $n$ be a positive integer. We consider a lexicographic product of the infinite path graph and a graph $G$ that can be either the complete or empty graph on $n$ vertices. We give a complete description of perfect colorings with an arbitrary number of colors of such graph products.
We prove that for a line perfect multigraph the chromatic index is equal to the list chromatic index. This is a generalization of Galvin's result on bipartite multigraphs. Soon after the first version was submitted to arxiv, I found out that the same result has already been achieved in an older paper by Peterson and Woodall, by a similar but not entirely the same method.
A coloring of the $q$-ary $n$-dimensional cube (hypercube) is called perfect if, for every $n$-tuple $x$, the collection of the colors of the neighbors of $x$ depends only on the color of $x$. A Boolean-valued function is called correlation-immune of degree $n-m$ if it takes the value 1 the same number of times for each $m$-dimensional face of the hypercube. Let $f=χ^S$ be a characteristic function of some subset $S$ of hypercube. In the present paper it is proven the inequality $ρ(S)q({\rm cor}(f)+1)\leq α(S)$, where ${\rm cor}(f)$ is the maximum degree of the correlation immunity of $f$, $α(…
A coloring of the Boolean $n$-cube is called perfect if, for every vertex $x$, the collection of the colors of the neighbors of $x$ depends only on the color of $x$. A Boolean function is called correlation-immune of degree $n-m$ if it takes the value 1 the same number of times for each $m$-face of the Boolean $n$-cube. In the present paper it is proven that each Boolean function $χ^S$ ($S\subset E^n$) satisfies the inequality $${\rm nei}(S)+ 2({\rm cor}(S)+1)(1-ρ(S))\leq n,$$ where ${\rm cor}(S)$ is the maximum degree of the correlation immunity of $χ^S$, ${\rm nei} (S)= \frac{1}{|S|}\sum\lim…
We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such that every connected induced subgraph $H$ of $G$ uses exactly $ω(H)$ many colors where $ω(H)$ is the clique number of $H$. A graph is perfectly colorable if it admits a perfect coloring. We show that the class of perfectly colorable graphs is exactly the class of perfect paw-free graphs. It follows that perfectly colorable graphs can be recognized and colored in linear time.
Following Penrose, we introduce a family of graph functions defined in terms of contractions of certain products of symmetric tensors along the edges of a graph. Special cases of these functions enumerate edge colorings and cycles of arbitrary length in graphs (in particular, Hamiltonian cycles).
We study forced periodicity of two-dimensional configurations under certain constraints and use an algebraic approach to multidimensional symbolic dynamics in which $d$-dimensional configurations and finite patterns are presented as formal power series and Laurent polynomials, respectively, in $d$ variables. We consider perfect colorings that are configurations such that the number of points of a given color in the neighborhood of any point depends only on the color of the point for some fixed relative neighborhood, and we show that by choosing the alphabet suitably any perfect coloring has a …
Consider a graph $G$ with a coloring of its edge set $E(G)$ from a set $Q = \set{c_1,c_2, \ldots, c_q}$. Let $Q_i$ be the set of all edges colored with $c_i$. Recently, Frieze defined a notion of the perfect matching color profile denoted by $\mcp(G)$, which is the set of vectors $(m_1, m_2, \ldots, m_q) \in [n]^q$ such that there exists a perfect matching $M$ in $G$ with $|Q_i \cap M| = m_i$ for all $i$. Let $\a_1, \a_2, \ldots, \a_q$ be positive constants such that $\sum_{i=1}^q \a_i = 1$. Let $G$ be the random bipartite graph $G_{n,n,p}$. Suppose the edges of $G$ are independently colored w…
We construct perfect $2$-colorings of the $12$-hypercube that attain our recent bound on the dimension of arbitrary correlation immune functions. We prove that such colorings with parameters $(x,12-x,4+x,8-x$) exist if $x=0$, $2$, $3$ and do not exist if $x=1$. This is a translation into English of the original paper by D. G. Fon-Der-Flaass, "Perfect colorings of the $12$-cube that attain the bound on correlation immunity", published in Russian in Siberian Electronic Mathematical Reports, vol. 4 (2007) 292-295.
We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variables agrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloring is a special case of the scheduling problems defined by Breuer and Klivans. We show ho…
We study a combinatorial coloring game between two players, Spoiler and Algorithm, who alternate turns. First, Spoiler places a new token at a vertex in $G$, and Algorithm responds by assigning a color to the new token. Algorithm must ensure that tokens on the same or adjacent vertices receive distinct colors. Spoiler must ensure that the token graph (in which two tokens are adjacent if and only if their distance in $G$ is at most $1$) has chromatic number at most $w$. Algorithm wants to minimize the number of colors used, and Spoiler wants to force as many colors as possible. Let $f(w,G)$ be …
A perfect structure is a triple $(M,P,S)$ of matrices $M, P$ and $S$ of consistent sizes such that $MP = PS$. Perfect structures comprise similar matrices, eigenvectors, perfect colorings (equitable partitions) and graph coverings. In this paper we study general algebraic properties of perfect structures and characterize all perfect structures with identity or unity matrix $M$. Next, we consider a graph product generalizing most standard products (e.g. Cartesian, tensor, normal, lexicographic graph products). For this product we propose a construction of perfect structures and prove that it ca…
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…
A vertex coloring of a graph is called "perfect" if for any two colors $a$ and $b$, the number of the color-$b$ neighbors of a color-$a$ vertex $x$ does not depend on the choice of $x$, that is, depends only on $a$ and $b$ (the corresponding partition of the vertex set is known as "equitable"). A set of vertices is called "completely regular" if the coloring according to the distance from this set is perfect. By the "weight distribution" of some coloring with respect to some set we mean the information about the number of vertices of every color at every distance from the set. We study the wei…
Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by $Γ^{\prime}(R)$, is a graph with vertices in $W^*(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct vertices $a$ and $b$ in $W^*(R)$ are adjacent if and only if $a\not\in Rb$ and $b\not\in Ra$. In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.
There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R/J^s, for any power J^s of J, in terms of the coloring properties of hypergraphs arising from H. We also give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J. Our work yields two new purely al…
We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset.
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…
In a recent paper by the same authors, we constructed a stationary 1-dependent 4-coloring of the integers that is invariant under permutations of the colors. This was the first stationary k-dependent q-coloring for any k and q. When the analogous construction is carried out for q>4 colors, the resulting process is not k-dependent for any k. We construct here a process that is symmetric in the colors and 1-dependent for every q>=4. The construction uses a recursion involving Chebyshev polynomials evaluated at $\sqrt{q}/2$.
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…
This work is the PhD thesis of Nicolas Trotignon, written in 2004 under the supervision of Frédéric Maffray. It is motivated by the desire for a better understanding of perfect graphs. The proof of the Claude Berge's perfect graph conjecture in 2002 by Chudnovsky, Robertson, Seymour and Thomas has shed a new light on this field of combinatorics. But some questions are still unsettled, particulary the existence of a combinatorial algorithm for the coloring of perfect graphs. An even pair of a graph is a pair of vertices such that every path joining them has even length. As proved by Fonlupt and…
The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynom…
A planar graph can be embedded in a piecewise linear manifold, and the lattice on each linear piece can be colored with 3-coloring. If a planar graph can be colored with multiple 3-coloring, i.e. coloring the graph in pieces with different 3-color subsets of 4 colors, then the graph is 4-colorable. In this paper, multiple 3-coloring was introduced, and then the combination and partition of planar graphs for multiple 3-coloring was studied. The study reveals that planar graphs can generally be decomposed into independent subgraphs, and each subgraph can be triangulated into a symmetric structur…
For a graph $G$, $χ(G)$ will denote its chromatic number, and $ω(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$. An integer-valued function $f$ is called a $χ$-binding function for a hereditary class of graphs $\cal C$ if $χ(G) \leq f(ω(G))$ for every graph $G\in \cal C$. The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. The problem of finding a polynomial $χ$-binding function for t…
We present a randomized algorithm which takes as input an undirected graph $G$ on $n$ vertices with maximum degree $Δ$, and a number of colors $k \geq (8/3 + o_Δ(1))Δ$, and returns -- in expected time $\tilde{O}(nΔ^{2}\log{k})$ -- a proper $k$-coloring of $G$ distributed perfectly uniformly on the set of all proper $k$-colorings of $G$. Notably, our sampler breaks the barrier at $k = 3Δ$ encountered in recent work of Bhandari and Chakraborty [STOC 2020]. We also sketch how to modify our methods to relax the restriction on $k$ to $k \geq (8/3 - ε_0)Δ$ for an absolute constant $ε_0 > 0$. As in t…
Polypolyhedra (after R. Lang) are compounds of edge-transitive 1-skeleta. There are 54 topologically different polypolyhedra, and each has icosidodecahedral, cuboctahedral, or tetrahedral symmetry, all are realizable as modular origami models with one module per skeleton edge. Consider a coloring in which each edge of a given component receives a different color, and where the coloring (up to global color permutation) is invariant under the polypolyhedron's symmetry group. On the Five Intersecting Tetrahedra, the edges of each color form visual bands on the model, and correspond to matchings o…
If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger ($P_{10}$-conjecture) states that for any bridgeless cubic graph $G$, one has: $P_{10}\prec G$. The Sylvester coloring conjecture ($S_{10}$-conjecture) states that for any cubic graph $G$, $S_{10}\prec G$. In this paper, we introduce two new conjectures that are related to t…
In this we consider weighted symmetric digraph. Our result generalizes the work of Zhu (J.Comb.Theory, Ser.B, 86 (2002) 109-113) concerning the (k,d)-coloring of a graph, and thus is also a generalization of a corresponding result of Tuza (J.Comb.Theory, Ser.B, 55 (1992) 236-243). Our result also strengthens a result of Goddyn, Tarsi and Zhang (J.Graph Theory 28 (1998) 155-161) concerning the relation between orientation and the (k,d)-coloring of a graph.
