In this paper, we settle the open complexity status of interval constrained coloring with a fixed number of colors. We prove that the problem is already NP-complete if the number of different colors is 3. Previously, it has only been known that it is NP-complete, if the number of colors is part of the input and that the problem is solvable in polynomial time, if the number of colors is at most 2. We also show that it is hard to satisfy almost all of the constraints for a feasible instance.
An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an interval $t$-coloring if for each $i\in \{1,2,\ldots,t\}$ there is at least one edge of $G$ colored by $i$, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In this paper we prove that if a connected graph $G$ with $n$ vertices admits an interval $t$-coloring, then $t\leq 2n-3$. We also show that if $G$ is a connected $r$-regular graph with $n$ vertices has an interval $t$-coloring and $n\geq 2r+2$, then this upper bound can be improved to $2n-5$.
An interval edge t-coloring of a graph G is a proper edge coloring of G with colors 1,2...,t such that at least one edge of G is colored by color i,i=1,2...,t, and the edges incident with each vertex x are colored by d_{G}(x) consecutive colors, where d_{G}(x) is the degree of the vertex x in G. For Mobius ladders the existence of this coloring is proved and all possible numbers of colors in such colorings are found.
A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An \emph{interval total $t$-coloring} of a graph $G$ is a total coloring of $G$ with colors $1,2,\...,t$ such that at least one vertex or edge of $G$ is colored by $i$, $i=1,2,\...,t$, and the edges incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this paper we investigate some properties of interval total colorings. We also…
An interval edge t-coloring of a graph G is a proper edge coloring of G with colors 1,2...,t such that at least one edge of G is colored by color i,i=1,2...,t, and the edges incident with each vertex v are colored by d_{G}(v) consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In this paper interval edge colorings of bipartite cylinders and bipartite tori are investigated.
An edge coloring of a graph $G$ with colors $1,2,..., t$ is called an interval $t$-coloring if for each $i\in \{1,2,...,t\}$ there is at least one edge of $G$ colored by $i$, the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In 1994 Asratian and Kamalian proved that if a connected graph $G$ admits an interval $t$-coloring, then $t\leq (d+1) (Δ-1) +1$, and if $G$ is also bipartite, then this upper bound can be improved to $t\leq d(Δ-1) +1$, where $Δ$ is the maximum degree in $G$ and $d$ is the diameter of $G$. In this paper we show that these upper…
An edge coloring of a graph $G$ with colors $1,2,\ldots ,t$ is called an interval $t$-coloring if for each $i\in \{1,2,\ldots,t\}$ there is at least one edge of $G$ colored by $i$, 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 there is an integer $t\geq 1$ for which $G$ has an interval $t$-coloring. Let $\mathfrak{N}$ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if $G,H\in \mathfrak{N}$, then the Cartesian product of these graphs belongs to $\mathfrak{N}$. Also, the…
We introduce a generalization of the well known graph (vertex) coloring problem, which we call the problem of \emph{component coloring of graphs}. Given a graph, the problem is to color the vertices using minimum number of colors so that the size of each connected component of the subgraph induced by the vertices of the same color does not exceed $C$. We give a linear time algorithm for the problem on proper interval graphs. We extend this algorithm to solve two weighted versions of the problem in which vertices have integer weights. In the \emph{splittable} version the weights of vertices can…
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{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. In 1990, Kamalian proved that if a graph $G$ with at least one edge has an interval $t$-coloring, then $t\leq 2|V(G)|-3$. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph $G$ admits an interval $t$-coloring, then $t\leq \frac{11}{6}|V(G)|$. In the same paper Axenovich suggested a conjecture that if a planar graph $G$ has an interval $t$-colo…
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ 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 it has an interval $t$-coloring for some positive integer $t$. The problem of deciding whether a bipartite graph is interval colorable is NP-complete. The smallest known examples of interval non-colorable bipartite graphs have $19$ vertices. On the other hand it is known that the bipartite graphs on at most $14$ vertices are interval colorable. In th…
A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that each of $t$ colors is used, and adjacent edges are colored differently. The set of colors of edges incident with a vertex $x$ of $G$ is called a spectrum of $x$. A proper edge $t$-coloring of a graph $G$ is interval for its vertex $x$ if the spectrum of $x$ is an interval of integers. A proper edge $t$-coloring of a graph $G$ is persistent-interval for its vertex $x$ if the spectrum of $x$ is an interval of integers beginning from the color 1. For graphs $G$ from some classes of graphs, we…
A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An interval total $t$-coloring of a graph $G$ is a total coloring of $G$ with colors $1,\ldots,t$ such that all colors are used, and the edges incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. In this paper we prove that all complete multipartite graphs with the same number of vertices in each part are interval total colorable. Mo…
An edge-coloring of a multigraph G with colors 1,2,...,t is called an 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. In this paper we prove that if G is a connected cubic multigraph (a connected cubic graph) that admits an interval t-coloring, then t\leq |V(G)| +1 (t\leq |V(G)|), where V(G) is the set of vertices of G. Moreover, if G is a connected cubic graph, G\neq K_{4}, and G has an interval t-coloring, then t\leq |V(G)| -1. We also show that these upper bounds are sharp. Finally, we prove that …
An equitable tree-$k$-coloring of a graph is a vertex $k$-coloring such that each color class induces a forest and the size of any two color classes differ by at most one. In this work, we show that every interval graph $G$ has an equitable tree-$k$-coloring for any integer $k\geq \lceil(Δ(G)+1)/2\rceil$, solving a conjecture of Wu, Zhang and Li (2013) for interval graphs, and furthermore, give a linear-time algorithm for determining whether a proper interval graph admits an equitable tree-$k$-coloring for a given integer $k$. For disjoint union of split graphs, or $K_{1,r}$-free interval grap…
An interval t-coloring of a graph G is a proper edge-coloring of G with colors 1,2,...,t such that at least one edge of G is colored by i, i=1,2,...,t, and the edges incident to each vertex v\in V(G) are colored by d_{G}(v) consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In this paper interval edge-colorings of various graph products are investigated.
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an 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 $t$. For an interval colorable graph $G$, the least value of $t$ for which $G$ has an interval $t$-coloring is denoted by $w(G)$. A graph $G$ is outerplanar if it can be embedded in the plane so that all its vertices lie on the same (unbounded) face. In this paper we show that if $G$ is …
In this note we prove that K_{1,m,n} is interval edge-colorable if and only if gcd(m+1,n+1)=1. It settles in the affirmative a conjecture of Petrosyan.
A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $v\in V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo $t$, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. A graph $G$ is \emph{interval cyclically colorable} if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by $\mathfrak{N}_{c}$. For a graph $G\in \mathfrak{N}_{c}$, the least and the greatest values of $t$ for which it …
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…
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…
A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex $x$ is called a spectrum of $x$. An arbitrary nonempty subset of consecutive integers is called an interval. We say that a proper edge $t$-coloring of a graph $G$ is interval in the vertex $x$ if the spectrum of $x$ is an interval. We say that a proper edge $t$-coloring $φ$ of a graph $G$ is interval on a subset $R_0$ of vertices of $G$, if for an arbitrary $x\in R_0$, $…
We consider proper online colorings of hypergraphs defined by geometric regions. We prove that there is an online coloring algorithm that colors $N$ intervals of the real line using $Θ(\log N/k)$ colors such that for every point $p$, contained in at least $k$ intervals, not all the intervals containing $p$ have the same color. We also prove the corresponding result about online coloring a family of wedges (quadrants) in the plane that are the translates of a given fixed wedge. These results contrast the results of the first and third author showing that in the quasi-online setting 12 colors ar…
An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ 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 it has an interval $t$-coloring for some positive integer $t$. For an interval colorable graph $G$, $W(G)$ denotes the greatest value of $t$ for which $G$ has an interval $t$-coloring. It is known that the complete graph is interval colorable if and only if the number of its vertices is even. However, the exact value of $W(K_{2n})$ is known only fo…
We consider the k-strong conflict-free coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring has to be conflict-free, in the sense that in every interval I there are at least k colors each appearing exactly once in I. In this paper, we present a polynomial algorithm for the general problem; the algorithm has an approximation factor 5-2/k when k\geq2 and approximation factor 2 for k=1. In the special case the family contains all the possible intervals on the given set of points, we show that a 2 approxim…
A graph $G$ is called a complete $k$-partite ($k\geq 2$) graph if its vertices can be partitioned into $k$ independent sets $V_{1},...,V_{k}$ such that each vertex in $V_{i}$ is adjacent to all the other vertices in $V_{j}$ for $1\leq i<j\leq k$. A complete $k$-partite graph $G$ is a complete balanced $k$-partite graph if $|V_{1}| = |V_{2}| =... = |V_{k}|$. 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 colorabl…
A translation from Russian of the work of R.R. Kamalian "Interval colorings of complete bipartite graphs and trees", Preprint of the Computing Centre of the Academy of Sciences of Armenia, Yerevan, 1989. (Was published by the decision of the Academic Council of the Computing Centre of the Academy of Sciences of Armenian SSR and Yerevan State University from 7.09.1989).
For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $φ:E(G)\rightarrow\{1,2,\ldots,t\}$ is called a proper edge $t$-coloring of a graph $G$ if adjacent edges are colored differently and each of $t$ colors is used. An arbitrary nonempty subset of consecutive integers is called an interval. If $φ$ is a proper edge $t$-coloring of a graph $G$ and $x\in V(G)$, then $S_G(x,φ)$ denotes the set of colors of edges of $G$ which are incident with $x$. A proper edge $t$-coloring $φ$ of a graph $G$ is called a …
An edge-coloring of a graph $G$ with colors $1,...,t$ is called an 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. In 1991 Erdős constructed a bipartite graph with 27 vertices and maximum degree 13 which has no interval coloring. Erdős's counterexample is the smallest (in a sense of maximum degree) known bipartite graph which is not interval colorable. On the other hand, in 1992 Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this paper we give …
An edge-coloring of a multigraph $G$ with colors $1,\ldots,t$ is called an 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. In this note, we show that all Eulerian multigraphs with an odd number of edges have no interval coloring. We also give some methods for constructing of interval non-edge-colorable Eulerian multigraphs.
Let $G=(V_1(G),V_2(G),E(G))$ be a bipartite multigraph, and $R\subseteq V_1(G)\cup V_2(G)$. A proper coloring of edges of $G$ with the colors $1,\ldots,t$ is called interval (respectively, continuous) on $R$, if each color is used for at least one edge and the edges incident with each vertex $x\in R$ are colored by $d(x)$ consecutive colors (respectively, by the colors $1,\ldots,d(x))$, where $d(x)$ is a degree of the vertex $x$. We denote by $w_1(G)$ and $W_1(G)$, respectively, the least and the greatest values of $t$, for which there exists an interval on $V_1(G)$ coloring of the multigraph …
The flooding games, which are called Flood-It, Mad Virus, or HoneyBee, are a kind of coloring games and they have been becoming popular online. In these games, each player colors one specified cell in his/her turn, and all connected neighbor cells of the same color are also colored by the color. This flooding or coloring spreads on the same color cells. It is natural to consider these new coloring games on more general boards, or general graphs. Recently, computational complexities of the variants of the flooding games on several graph classes have been studied. In this paper, we investigate t…
A facial-parity edge-coloring of a $2$-edge-connected plane graph is a facially-proper edge-coloring in which every face is incident with zero or an odd number of edges of each color. A facial-parity vertex-coloring of a $2$-connected plane graph is a facially-proper vertex-coloring in which every face is incident with zero or an odd number of vertices of each color. Czap and Jendroľ (in Facially-constrained colorings of plane graphs: A survey, Discrete Math. 340 (2017), 2691--2703), conjectured that $10$ colors suffice in both colorings. We present an infinite family of counterexamples to bot…
For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $φ:E(G)\rightarrow\{1,2,\ldots,t\}$ is called a proper edge $t$-coloring of a graph $G$ if adjacent edges are colored differently and each of $t$ colors is used. An arbitrary nonempty subset of consecutive integers is called an interval. If $φ$ is a proper edge $t$-coloring of a graph $G$ and $x\in V(G)$, then $S_G(x,φ)$ denotes the set of colors of edges of $G$ which are incident with $x$. A proper edge $t$-coloring $φ$ of a graph $G$ is called a …
Given a hypergraph H = (V, E), a coloring of its vertices is said to be conflict-free if for every hyperedge S \in E there is at least one vertex in S whose color is distinct from the colors of all other vertices in S. The discrete interval hypergraph Hn is the hypergraph with vertex set {1,...,n} and hyperedge set the family of all subsets of consecutive integers in {1,...,n}. We provide a polynomial time algorithm for conflict-free coloring any subhypergraph of Hn, we show that the algorithm has approximation ratio 2, and we prove that our analysis is tight, i.e., there is a subhypergraph fo…
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an integer interval. It is well-known that there are graphs that do not have interval colorings. The \emph{deficiency} of a graph $G$, denoted by $\mathrm{def}(G)$, is the minimum number of pendant edges whose attachment to $G$ leads to a graph admitting an interval coloring. In this paper we investigate the problem of determining or bounding of the deficiency of complete multipartite graphs. In particul…
A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of $G$ is the maximum integer $χ_b(G)$ for which $G$ has a b-coloring with $χ_b(G)$ colors. A graph $G$ is b-continuous if $G$ has a b-coloring with $k$ colors, for every integer $k$ in the interval $[χ(G),χ_b(G)]$. It is known that not all graphs are b-continuous. Here, we investigate whether the lexicographic product $G[H]$ of b-continuous graphs $G$ and $H$ is also b-continuous. Using homomorphisms, we provide a new lower b…
For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $φ:E(G)\rightarrow \{1,...,t\}$ is called a proper edge $t$-coloring of a graph $G$, if adjacent edges are colored differently and each of $t$ colors is used. The least value of $t$ for which there exists a proper edge $t$-coloring of a graph $G$ is denoted by $χ'(G)$. For any graph $G$, and for any integer $t$ satisfying the inequality $χ'(G)\leq t\leq |E(G)|$, we denote by $α(G,t)$ the set of all proper edge $t$-colorings of $G$. Let us also defi…
It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces.
In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals is allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains $k$-colorable at all times, and the updates consist of insertions only, then we can achieve the amortized recourse budget of $O(…
The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction of triangle-free geometric intersection graphs with arbitrarily large chromatic number due to Pawlik et al. We show that on-line graph coloring problems give rise to classes of game graphs with a natural geometric interpretation. We use this concept to estimate the chromatic number of graphs with geometric representations by finding, for appropriate simpler g…
A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex $x$ is called a spectrum of $x$. An arbitrary nonempty subset of consecutive integers is called an interval. Suppose that all edges of a graph $G$ are colored in the game of Alice and Bob with asymmetric distribution of roles. Alice determines the number $t$ of colors in the future proper edge coloring of $G$ and aspires to minimize the number of vertices with an interva…
A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of $G$ is the maximum integer $b(G)$ for which $G$ has a b-coloring with $b(G)$ colors. A graph $G$ is b-continuous if $G$ has a b-coloring with $k$ colors, for every integer $k$ in the interval $[χ(G),b(G)]$. It is known that not all graphs are b-continuous. In this article, we show that if $G$ has girth at least 10, then $G$ is b-continuous.
A packing $k$-coloring for some integer $k$ of a graph $G=(V,E)$ is a mapping $φ:V\to\{1,\ldots,k\}$ such that any two vertices $u, v$ of color $φ(u)=φ(v)$ are in distance at least $φ(u)+1$. This concept is motivated by frequency assignment problems. The \emph{packing chromatic number} of $G$ is the smallest $k$ such that there exists a packing $k$-coloring of $G$. Fiala and Golovach showed that determining the packing chromatic number for chordal graphs is \NP-complete for diameter exactly 5. While the problem is easy to solve for diameter 2, we show \NP-completeness for any diameter at least…
Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $Θ(\log\log n)$. Essentially the same construction produces $Θ(\log\log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in $\mathbb{R}^2$ closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles…
We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n+k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k=0), we parameterize graph problems by k,…
We study the problem of non-preemptively scheduling $n$ jobs, each job $j$ with a release time $t_j$, a deadline $d_j$, and a processing time $p_j$, on $m$ parallel identical machines. Cieliebak et al. (2004) considered the two constraints $|d_j-t_j|\leq λp_j$ and $|d_j-t_j|\leq p_j +σ$ and showed the problem to be NP-hard for any $λ>1$ and for any $σ\geq 2$. We complement their results by parameterized complexity studies: we show that, for any $λ>1$, the problem remains weakly NP-hard even for $m=2$ and strongly W[1]-hard parameterized by $m$. We present a pseudo-polynomial-time algorithm for…
We consider an online vector balancing question where $T$ vectors, chosen from an arbitrary distribution over $[-1,1]^n$, arrive one-by-one and must be immediately given a $\pm$ sign. The goal is to keep the discrepancy small as possible. A concrete example is the online interval discrepancy problem where T points are sampled uniformly in [0,1], and the goal is to immediately color them $\pm$ such that every sub-interval remains nearly balanced. As random coloring incurs $Ω(T^{1/2})$ discrepancy, while the offline bounds are $Θ(\sqrt{n \log (T/n)})$ for vector balancing and $1$ for interval ba…
We provide new (parameterized) computational hardness results for Interval Scheduling on Unrelated Machines. It is a classical scheduling problem motivated from just-in-time or lean manufacturing, where the goal is to complete jobs exactly at their deadline. We are given $n$ jobs and $m$ machines. Each job has a deadline, a weight, and a processing time that may be different on each machine. The goal is find a schedule that maximized the total weight of jobs completed exactly at their deadline. Note that this uniquely defines a processing time interval for each job on each machine. Interval Sc…
