We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in $[1,σ]$. For $σ=1$ it is the class of unit interval graphs, and for $σ=\infty$ the class of all interval graphs. Our focus is on intermediary classes.
We present a $(1+σ)$-competitive algorithm, which beats the state of the art for $1 < σ< 2$, and proves that the problem we study can be strictly easier than online coloring of general interval graphs.
On the lower bound side, we prove that no algorithm is better than $5/3$-competitive for any $σ>1$, nor better than $7/4$-competitive for any $σ>2$, and that no algorithm beats the $5/2$ asymptotic competitive ratio for all, arbitrarily large, values of $σ$. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than $2$.
In Raman (2007), using a column construction technique it is proved that every interval graph can be colored online with First Fit with at most $8w(G)$ colors, where $w(G)$ is the size of the maximum clique of $G$. Since the column construction can not be adapted to circular arc graphs we give a different proof to establish an upper bound of $9w(G)$ for online coloring a circular arc graph $G$ with the First Fit algorithm.
The on-line interval coloring and its variants are important combinatorial problems with many applications in network multiplexing, resource allocation and job scheduling. In this paper we present a new lower bound of $4.1626$ for the competitive ratio for the on-line coloring of intervals with bandwidth which improves the best known lower bound of $\frac{24}{7}$. For the on-line coloring of unit intervals with bandwidth we improve the lower bound of $1.831$ to $2$.
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 …
We propose two strategies for Presenter in the on-line interval graph coloring games. Specifically, we consider a setting in which each interval is associated with a $d$-dimensional vector of weights and the coloring needs to satisfy the $d$-dimensional bandwidth constraint, and the $k$-cardinality constraint. Such a variant was first introduced by Epstein and Levy and it is a natural model for resource-aware task scheduling with $d$ different shared resources where at most $k$ tasks can be scheduled simultaneously on a single machine. The first strategy forces any on-line interval coloring al…
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…
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 …
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 $1,...,t$ 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 $x\in R_0$, $…
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…
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 …
The chromatic number of signed graphs is defined recently. The coloring and clique problem of interval graphs has been studied and polynomial time algorithms are established. Here we consider these problems for signed interval graphs and prove that the coloring problem of signed interval graphs is NP-complete whereas their ordinary clique problem is in P. We also study the complexity of further related problems.
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.
We confirm the equitable $Δ$-coloring conjecture for interval graphs and establish the monotonicity of equitable colorability for them. We further obtain results on equitable colorability about square (or Cartesian) and cross (or direct) products of graphs.
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 …
An \emph{interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A \emph{cyclic interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. Denote by $w(G)$ ($w_{c}(G)$) and $W(G)$ ($W_{c}(G)$) the minimum and maximum number of colors in a (cyclic) i…
A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a \emph{cyclic interval $t$-coloring} if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. We prove that a bipartite graph $G$ with even maximum degree $Δ(G)\geq 4$ admits a cyclic interval $Δ(G)$-coloring if for every vertex $v$ the degree $d_G(v)$ satisfies either $d_G(v)\geq Δ(G)-2$ or $d_G(v)\leq 2$. We also prove that every Eulerian bipartite graph $G$ with maximum degree at most $8$ has a cyclic inter…
An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2,4,6,8}. We provide sufficient conditions for the …
We prove that it is always possible to color online nonrepetitively any (partial) $k$-tree (that is, graphs with tree-width at most $k$) with $4^k$ colors. This implies that it is always possible to color online nonrepetitively cycles, trees and series-parallel graphs with $16$ colors. Our results generalize the respective (offline) nonrepetitive coloring results.
We study the problem of consistent and homogeneous colourings for increasing families of dyadic intervals. We determine when this problem can be solved and when not.
In this paper, we give a family of online algorithms for the classical coloring problem of intersection graphs of discs with bounded diameter. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from using two techniques of partitioning the set of vertices before coloring them. One of whichis an application of a b-fold coloring of the plane. The method is more general and we show how it can be applied to coloring other shapes on the plane as well as adjust it for online …
A $k$-improper edge coloring of a graph $G$ is a mapping $α:E(G)\longrightarrow \mathbb{N}$ such that at most $k$ edges of $G$ with a common endpoint have the same color. An improper edge coloring of a graph $G$ is called an improper interval edge coloring if the colors of the edges incident to each vertex of $G$ form an integral interval. In this paper we introduce and investigate a new notion, the interval coloring impropriety (or just impropriety) of a graph $G$ defined as the smallest $k$ such that $G$ has a $k$-improper interval edge coloring; we denote the smallest such $k$ by $μ_{\mathr…
For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $χ'_{int}(G,X)$ the minimum $k$ such that $G$ has an $X$-interval coloring with $k$ colors. The author and Toft conjectured [Discrete Mathematics 339 (2016), 2628--2639] that there is a polynomial $P(x)$ such that if $G$ has maximum degree at most $Δ$, then $χ'_{int}(G,X) \leq P(Δ)$. In this short note, we prove this conjecture; in fact, we prove that a cubic polynomial suffices. We al…
We give an upper bound for the online chromatic number of graphs with high girth and for graphs with high oddgirth generalizing Kier- stead's algorithm for graphs that contain neither a C3 or C5 as an induced subgraph.
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 …
The online list coloring is a widely studied topic in graph theory. A graph $G$ is 2-paintable if we always have a strategy to complete a coloring in an online list coloring of $G$ in which each vertex has a color list of size 2. In this paper, we focus on the online list coloring game in which the number of colors is known in advance. We say that $G$ is $[2,t]$-paintable if we always have a strategy to complete a coloring in an online list coloring of $G$ in which we know that there are exactly $t$ colors in advance, and each vertex has a color list of size 2. Let $M(G)$ denote the maximum $t…
A proper edge-coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval of consecutive integers. Interval thickness s(G) of a graph G is the smallest number of interval colorable graphs edge-decomposing G. We prove that s(G)=o(n) for any graph G on n vertices. This improves the previously known bound of 2n/5 by Asratian, Casselgren, and Petrosyan. While we do not have a single example of a graph with interval thickness strictly greater than 2, we construct bipartite graphs whose interval spectrum has arbitrarily many arbitrarily large gaps. …
Let D be a finite set of positive real numbers. The distance graph G(R,D) is the graph with vertex set R (set of real numbers), and two vertices x, y are adjacent if |x-y| belongs to D. We prove that every positive integer t>1 there is a distance set D such that the chromatic number of G(R,D) is t and no proper coloring of G(R,D) with t colors allows monochromatic intervals. This result disproves a conjecture in [2].
First-fit is the online graph coloring algorithm that considers vertices one at a time in some order and assigns each vertex the least positive integer not used already on a neighbor. The maximum number of colors used by first-fit on graph G over all vertex orders is denoted χ_{FF}(G). The exact value of R := \sup_G [χ_{FF}(G) / ω(G)] over interval graphs G is unknown. Pemmaraju, Raman, and Varadarajan (2004) proved R <= 10, and this can be improved to 8. Witsenhausen (1976) and Chrobak and Ślusarek (1988) showed R >= 4, and Ślusarek (1993) improved this to 4.45. We prove R >= 5.
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…
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 …
In the mysterious and colorful world of chromatic numbers, where there are a lot of unknown, there is an amazing thing. It turns out that for some intervals of forbidden distances on the plane, one can specify the exact value of the chromatic number $χ$. Two sets of such intervals have been found, for $χ=7$ and 9. We call them islands of certainty. Here we increase the size of these islands, and add three new ones with $χ=8$, 12, 13. We also %formulate conjectures which predict conjecture islands for $χ=14$, 15, 16. Are there islands of certainty for $χ$=10 or 11? This is still a mystery. Roll…
A \textit{distinguishing coloring} of a graph $G$ is a coloring of the vertices so that every nontrivial automorphism of $G$ maps some vertex to a vertex with a different color. The \textit{distinguishing number} of $G$ is the minimum $k$ such that $G$ has a distinguishing coloring where each vertex is assigned a color from $\{1,\ldots,k\}$. A \textit{list assignment} to $G$ is an assignment $L=\{L(v)\}_{v\in V(G)}$ of lists of colors to the vertices of $G$. A \textit{distinguishing $L$-coloring} of $G$ is a distinguishing coloring of $G$ where the color of each vertex $v$ comes from $L(v)$. T…
We take an application of the Kernel Lemma by Kostochka and Yancey to its logical conclusion. The consequence is a sort of magical way to draw conclusions about list coloring (and online list coloring) just from the existence of an independent set incident to many edges. We use this to prove an Ore-degree version of Brooks' Theorem for online list-coloring. The Ore-degree of an edge $xy$ in a graph $G$ is $θ(xy) = d_G(x) + d_G(y)$. The Ore-degree of $G$ is $θ(G) = \max_{xy\in E(G)}θ(xy)$. We show that every graph with $θ\ge18$ and $ω\le\fracθ{2}$ is online $\left\lfloor \fracθ{2}\right\rfloor …
The DP-coloring problem is a generalization of the list-coloring problem in which the goal is to find an independent transversal in a certain topological cover of a graph $G$. In the online DP-coloring problem, the cover of $G$ is revealed one component at a time, and the independent transversal of the cover must be constructed in parts based on incomplete information. Kim, Kostochka, Li, and Zhu asked whether the chromatic numbers corresponding to these two graph coloring problems can have an arbitrarily large difference in a single graph. We answer this question in the affirmative by constru…
The slow-coloring game is played by Lister and Painter on a graph $G$. On each round, Lister marks a nonempty subset $M$ of the uncolored vertices, scoring $|M|$ points. Painter then gives a color to a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. Painter and Lister want to minimize and maximize the total score, respectively. The best score that each player can guarantee is the sum-color cost of $G$, written $\mathring{\mathrm{s}}(G)$. The game is an online variant of online sum list coloring. We proe $\frac{|V(G)|}{2α(G)} + \frac{1}{2} \leq \frac{\math…
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(…
We introduce the concept of low rank-width colorings, generalising the notion of low tree-depth colorings introduced by Nešetřil and Ossona de Mendez in [Grad and classes with bounded expansion I. Decompositions. EJC, 2008]. We say that a class $\mathcal{C}$ of graphs admits low rank-width colourings if there exist functions $N\colon \mathbb{N}\rightarrow\mathbb{N}$ and $Q\colon \mathbb{N}\rightarrow\mathbb{N}$ such that for all $p\in \mathbb{N}$, every graph $G\in \mathcal{C}$ can be vertex colored with at most $N(p)$ colors such that the union of any $i\leq p$ color classes induces a subgrap…
