Let $H$ be a graph with $Δ(H) \leq 2$, and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. We prove that if $H$ contains at most one odd cycle of length exceeding $3$, or if $H$ contains at most $3$ triangles, then $χ(G) \leq 4$. This proves the Strong Coloring Conjecture for such graphs $H$. For graphs $H$ with $Δ=2$ that are not covered by our theorem, we prove an approximation result towards the conjecture.
DP-coloring was introduced by Dvořák and Postle as a generalization of list coloring. It was originally used to solve a longstanding conjecture by Borodin, stating that every planar graph without cycles of lengths 4 to 8 is 3-choosable. In this paper, we give three sufficient conditions for a planar graph to be DP-4-colorable. Actually all the results (Theorem 1.3, 1.4 and 1.7) are stated in the ``color extendability'' form, and uniformly proved by vertex identification and discharging method.
A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and Göring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4…
Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a $(Δ(G)+1)$-edge coloring where $Δ(G)$ denotes the maximum degreee of $G$. Recently, Cabello raised the following question: given two graphs $G_1,G_2$ of maximum degree $Δ$ on the same set of vertices $V$, is it possible to edge-color their (edge) union with $Δ+2$ colors in such a way the restriction of $G$ to respectively the edges of $G_1$ and the edges of $G_2$ …
The paper characterizes $(2t+1)$-regular graphs with circular flow number $2 + \frac{2}{2t-1}$. For $t=1$ this is Tutte's characterization of cubic graphs with flow number 4. The class of cubic graphs is the only class of odd regular graphs where a flow number separates the class 1 graphs from the class 2 graphs. We finally state some conjectures and relate them to existing flow-conjectures.
An edge-coloring of a graph $G$ with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of $G$ are distinct and the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum of a graph $G$ is the sum of the colors of edges in a sum edge-coloring of $G$. It is known that the problem of finding the edge-chromatic sum of an $r$-regular ($r\geq 3$) graph is $NP$-complete. In this paper we give a polynomial time $(1+\frac{2r}{(r+1)^{2}})$-approximation algorithm for the edge-chromatic sum problem on $r$-regular graphs for $r\geq 3$. Also, it i…
Let $\binom{X}{h}$ be the collection of all $h$-subsets of an $n$-set $X\supseteq Y$. Given a coloring (partition) of a set $S\subseteq \binom{X}{h}$, we are interested in finding conditions under which this coloring is extendible to a coloring of $\binom{X}{h}$ so that the number of times each element of $X$ appears in each color class (all sets of the same color) is the same number $r$. The case $S=\varnothing, r=1$ was studied by Sylvester in the 18th century, and remained open until the 1970s. The case $h=2,r=1$ is extensively studied in the literature and is closely related to completing …
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.
An $(r-1,1)$-coloring of an $r$-regular graph $G$ is an edge coloring such that each vertex is incident to $r-1$ edges of one color and $1$ edge of a different color. In this paper, we completely characterize all $4$-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a $(3,1)$-coloring. An $\{r-1,1\}$-factor of an $r$-regular graph is a spanning subgraph in which each vertex has degree either $r-1$ or $1$. We prove various conditions that that must hold for any vertex-minimal $5$-regular pseudographs without $(4,1)$-colorings or without $\{4,1\}$-factors.…
A {\em total coloring} of a graph $G$ is an assignment of colors to the vertices and the edges of $G$ such that every pair of adjacent/incident elements receive distinct colors. The {\em total chromatic number} of a graph $G$, denoted by $\chiup''(G)$, is the minimum number of colors in a total coloring of $G$. The well-known Total Coloring Conjecture (TCC) says that every graph with maximum degree $Δ$ admits a total coloring with at most $Δ+ 2$ colors. A graph is {\em $1$-toroidal} if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate t…
Let $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this note, we show that if $G$ is a graph with $Δ(G)-δ(G)\leq 1$ and $χ^{\prime}(G)=Δ(G)=r$ ($r\geq 3$), then $G$ has an $R$-sequential $r$-coloring with $\vert R\vert \geq \left\lceil\frac{(r-1)n_{r}+n}{r}\right\rceil$, where $n=\vert V(G)\vert$ and $n_{r}=\vert\{v\in V(G):d_{G}(v)=r\}\vert$. As a c…
We consider Kempe changes on the $k$-colorings of a graph on $n$ vertices. If the graph is $(k-1)$-degenerate, then all its $k$-colorings are equivalent up to Kempe changes. However, the sequence between two $k$-colorings that arises from the proof may be exponential in the number of vertices. An intriguing open question is whether it can be turned polynomial. We prove this to be possible under the stronger assumption that the graph has treewidth at most $k-1$. Namely, any two $k$-colorings are equivalent up to $O(kn^2)$ Kempe changes. We investigate other restrictions (list coloring, bounded …
At most how many (proper) q-colorings does a regular graph admit? Galvin and Tetali conjectured that among all n-vertex, d-regular graphs with 2d|n, none admits more q-colorings than the disjoint union of n/2d copies of the complete bipartite graph K_{d,d}. In this note we give asymptotic evidence for this conjecture, giving an upper bound on the number of proper q-colorings admitted by an n-vertex, d-regular graph of the form a^n b^{n(1+o(1))/d} (where a and b depend on q and where o(1) goes to 0 as d goes to infinity) that agrees up to the o(1) term with the count of q-colorings of n/2d copi…
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. Fiam${\rm \check{c}}$ik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that $a'(G)\le Δ+ 2$ for any simple graph $G$ with maximum degree $Δ$. Basavaraju and Chandran (2009) showed that every graph $G$ with $Δ=4$, which is not 4-regular, satisfies the conjecture. In this paper, we settle the 4-regular case, i.e., we show that every 4-regular g…
We study the \emph{maximum differential coloring problem}, where the vertices of an $n$-vertex graph must be labeled with distinct numbers ranging from $1$ to $n$, so that the minimum absolute difference between two labels of any two adjacent vertices is maximized. As the problem is \NPH for general graphs~\cite{leung1984}, we consider planar graphs and subclasses thereof. We initially prove that the maximum differential coloring problem remains \NPH, even for planar graphs. Then, we present tight bounds for regular caterpillars and spider graphs. Using these new bounds, we prove that the Mill…
Using a relation between domination number and incidence chromatic number, we obtain necessary and sufficient conditions for $r$-regular graphs to be $(r+1)$-incidence colorable. Also, we determine the optimal Nordhaus-Gaddum inequality for the incidence chromatic number.
In this article, we study subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and study Sturmian colorings, which are colorings of minimal unbounded subword complexity. We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray with loops possibly attached, thus a lifting of an "infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones.
If $G$ is a graph and $\mathcal{H}$ is a set of subgraphs of $G$, we say that an edge-coloring of $G$ is $\mathcal{H}$-polychromatic if every graph from $\mathcal{H}$ gets all colors present in $G$ on its edges. The $\mathcal{H}$-polychromatic number of $G$, denoted $\operatorname{poly}_\mathcal{H} (G)$, is the largest number of colors in an $\mathcal{H}$-polychromatic coloring. In this paper we determine $\operatorname{poly}_\mathcal{H} (G)$ exactly when $G$ is a complete graph on $n$ vertices, $q$ is a fixed nonnegative integer, and $\mathcal{H}$ is one of three families: the family of all m…
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…
An edge-coloring of a graph $G$ with consecutive integers $c_{1},\ldots,c_{t}$ is called an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The set of all interval colorable graphs is denoted by $\mathfrak{N}$. In 2004, Giaro and Kubale showed that if $G,H\in \mathfrak{N}$, then the Cartesian product of these graphs belongs to $\mathfrak{N}$. In the same year they formulated a similar proble…
We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi t…
A coloring of edges of a graph $G$ is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance $1$ in $G$ or in a common triangle. Naturally, the injective chromatic index of $G$, $χ'_{inj}(G)$, is the minimum number of colors needed for an injective edge-coloring of $G$. We study how large can be the injective chromatic index of $G$ in terms of maximum degree of $G$ when we have restrictions on girth and/or chromatic number of $G$. We also compare our bounds with analogous bounds on the strong chromatic index.
In this paper, we continue the study of $2$-colorings in hypergraphs. A hypergraph is $2$-colorable if there is a $2$-coloring of the vertices with no monochromatic hyperedge. It is known (see Thomassen [J. Amer. Math. Soc. 5 (1992), 217--229]) that every $4$-uniform $4$-regular hypergraph is $2$-colorable. Our main result in this paper is a strengthening of this result. For this purpose, we define a vertex in a hypergraph $H$ to be a free vertex in $H$ if we can $2$-color $V(H) \setminus \{v\}$ such that every hyperedge in $H$ contains vertices of both colors (where $v$ has no color). We prov…
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…
A simple but empirically efficient heuristic algorithm for the edge-coloring of graphs is presented. Its basic idea is the displacement of "conflicts" (repeated colors in the edges incident to a vertex) along paths of adjacent vertices whose incident edges are recolored by swapping alternating colors (that is, doing a Kempe interchange). The results of performance tests on random cubic and $Δ$-regular graphs are presented, and a full implementation of the algorithm is given to facilitate its use and the reproducibility of results.
Let $X,Y$ be finite sets, $r,s,h, λ\in \mathbb{N}$ with $s\geq r, X\subsetneq Y$. By $λ\binom{X}{h}$ we mean the collection of all $h$-subsets of $X$ where each subset occurs $λ$ times. A coloring of $λ\binom{X}{h}$ is {\it $r$-regular} if in every color class each element of $X$ occurs $r$ times. A one-regular color class is a {\it perfect matching}. We are interested in the necessary and sufficient conditions under which an $r$-regular coloring of $λ\binom{X}{h}$ can be embedded into an $s$-regular coloring of $λ\binom{Y}{h}$. Using algebraic techniques involving glueing together orbits of a…
For spin systems, such as the $q$-colorings and independent-set models, approximating the partition function in the so-called non-uniqueness region, where the model exhibits long-range correlations, is typically computationally hard for bounded-degree graphs. We present new algorithmic results for approximating the partition function and sampling from the Gibbs distribution for spin systems in the non-uniqueness region on random regular bipartite graphs. We give an $\mathsf{FPRAS}$ for counting $q$-colorings for even $q=O\big(\tfracΔ{\logΔ}\big)$ on almost every $Δ$-regular bipartite graph. Th…
A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant $c$ such that for every $k$-regular graph $G$, $χ_d(G)\leq χ(G)+ c\ln k +1$. Also, we introduce an upper bound for the dynamic list chromatic number of regular graphs.
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.
For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex graph with minimum degree $δ$ maximizes the number of $H$-colorings? We show that for non-regular $H$ and sufficiently large $n$, the complete bipartite graph $K_{δ,n-δ}$ is the unique maximizer. As a corollary, for non-regular $H$ and sufficiently large $n$ the graph $K_{k,n-k}$ is the unique $k$-connected graph that maximizes the number of $H$-colorings among …
This paper is concerned with symmetric $1$-dependent colorings of the $d$-ray star graph $\mathscr{S}^d$ for $d \ge 2$. We compute the critical point of the $1$-dependent hard-core processes on $\mathscr{S}^d$, which gives a lower bound for the number of colors needed for a $1$-dependent coloring of $\mathscr{S}^d$. We provide an explicit construction of a $1$-dependent $q$-coloring for any $q \ge 5$ of the infinite subgraph $\mathscr{S}^3_{(1,1,\infty)}$, which is symmetric in the colors and whose restriction to any path is some symmetric $1$-dependent $q$-coloring. We also prove that there i…
In 1973, Altshuler characterized the $6$-regular triangulations on the torus to be precisely those that are obtained from a regular triangulation of the $r \times s$ toroidal grid where the vertices in the first and last column are connected by a shift of $t$ vertices. Such a graph is denoted $T(r, s, t)$. In 1999, Collins and Hutchinson classified the $4$-colorable graphs $T(r, s, t)$ with $r, s \geq 3$. In this paper, we point out a gap in their classification and show how it can be fixed. Combined with the classification of the $4$-colorable graphs $T(1, s, t)$ by Yeh and Zhu in 2003, this …
A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph admits a strong edge coloring with $2k-1$ colors if and only if it covers the Kneser graph $K(2k-1,k-1)$. In particular, a cubic graph is strongly $5$-edge-colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. [Ars Combin. 29 B (1990), …
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks that for any simple and finite graph $G$, $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum degree of $G$. We prove the conjecture for connected graphs with $Δ(G) \le 4$, with the additional restriction that $m \le 2n-1$, where $n$ is the number of vertices and $m$ is the number of edges …
Let $G$ be a finite $d$-regular graph with a proper edge coloring. An edge Kempe switch is a new proper edge coloring of $G$ obtained by switching the two colors along some bi-chromatic cycle. We prove that any other edge coloring can be obtained by performing finitely many edge Kempe switches, provided that $G$ is replaced with a suitable finite covering graph. The required covering degree is bounded above by a constant depending only on $d$.
The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in the same part. If the restrictions are too harsh, then the topological Tverberg theorem fails. The colored Tverberg theorem corresponds to graphs constructed as disjoint unions of small complete graphs. Hell studied the case of paths and cycles. In graph theory these partitions are…
We prove a known 2-coloring of the integers $[N] := \{1,2,3,\ldots,N\}$ minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers that are all the same color. Previous work by Parrilo, Robertson and Saracino conjectured an optimal coloring for large $N$ that involves 12 colored blocks. Here, we prove that the conjecture is optimal among anti-symmetric colorings with 12 or fewer colored blocks. We leverage a connection to the coloring of the continuous interval $[0,1]$ used by Parrilo, R…
The regular graph of ideals of the commutative ring $R$, denoted by $Γ_{reg}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is shown that for every Artinian ring $R$, the edge chromatic number of $Γ_{reg}(R)$ equals its maximum degree. Then a formula for the clique number of $Γ_{reg}(R)$ is given. Also, it is proved that for every reduced ring $R$ with $n(\geq3)$ minimal prime ideals, the edge chromat…
An exact $(k,d)$-coloring of a graph $G$ is a coloring of its vertices with $k$ colors such that each vertex $v$ is adjacent to exactly $d$ vertices having the same color as $v$. The exact $d$-defective chromatic number, denoted $χ_d^=(G)$, is the minimum $k$ such that there exists an exact $(k,d)$-coloring of $G$. In an exact $(k,d)$-coloring, which for $d=0$ corresponds to a proper coloring, each color class induces a $d$-regular subgraph. We give basic properties for the parameter and determine its exact value for cycles, trees, and complete graphs. In addition, we establish bounds on $χ_d^…
Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring such that for every two distinct colors i, j at most one pair of adjacent vertices are colored with i and j. This type of coloring is edge-distinguishing and has potential applications in transportation network, computer network, airway network system. The results presented in this paper fall into two categories: in the first part of the paper we are concerne…
Let $G$ be a graph with no isolated vertices. A {\em $k$-coupon coloring} of $G$ is an assignment of colors from $[k] := \{1,2,\dots,k\}$ to the vertices of $G$ such that the neighborhood of every vertex of $G$ contains vertices of all colors from $[k]$. The maximum $k$ for which a $k$-coupon coloring exists is called the {\em coupon coloring number} of $G$, and is denoted $χ_{c}(G)$. In this paper, we prove that every $d$-regular graph $G$ has $χ_{c}(G) \geq (1 - o(1))d/\log d$ as $d \rightarrow \infty$, and the proportion of $d$-regular graphs $G$ for which $χ_c(G) \leq (1 + o(1))d/\log d$ t…
Coloring graphs is an important algorithmic problem in combinatorics with many applications in computer science. In this paper we study coloring tournaments. A chromatic number of a random tournament is of order $Ω(\frac{n}{\log(n)})$. The question arises whether the chromatic number can be proven to be smaller for more structured nontrivial classes of tournaments. We analyze the class of tournaments defined by a forbidden subtournament $H$. This paper gives a first quasi-polynomial algorithm running in time $e^{O(\log(n)^{2})}$ that constructs colorings of $H$-free tournaments using only $O(n…
For a fixed simple digraph $F$ and a given simple digraph $D$, an $F$-free $k$-coloring of $D$ is a vertex-coloring in which no induced copy of $F$ in $D$ is monochromatic. We study the complexity of deciding for fixed $F$ and $k$ whether a given simple digraph admits an $F$-free $k$-coloring. Our main focus is on the restriction of the problem to planar input digraphs, where it is only interesting to study the cases $k \in \{2,3\}$. From known results it follows that for every fixed digraph $F$ whose underlying graph is not a forest, every planar digraph $D$ admits an $F$-free $2$-coloring, a…
Let $c$ be an edge-coloring of the complete $n$-vertex graph $K_n$. The problem of finding properly colored and rainbow Hamilton cycles in $c$ was initiated in 1976 by Bollobás and Erd\H os and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek, Frieze and Ruciński. We generalize these results, giving sufficient local (resp. global) restrictions on the colorings which guarantee a properly colored (resp. rainbow) copy of a given hypergraph $G$. We also study multipartite analogues of these questions. We give (up to a constant factor) optimal suf…
