Interval Coloring Research Articles

Interval edge-colorings of composition of graphs

An edge-coloring of a graph G with consecutive integers c 1 , … , c 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 . The set of all interval colorable graphs is denoted by N . In 2004, Giaro and Kubale showed that if G , H ∈ N , then the Cartesian product of these graphs belongs to N . In the same year they formulated a similar problem for the composition of graphs as an open problem. Later, in 2009, the second author showed that if G , H ∈ N and H is a regular graph, then G [ H ] ∈ N . In this paper, we prove that if G ∈ N and H has an interval coloring of a special type, then G [ H ] ∈ N . Moreover, we show that all regular graphs, complete bipartite graphs and trees have such a special interval coloring. In particular, this implies that if G ∈ N and T is a tree, then G [ T ] ∈ N .

Interval edge-colorings of complete graphs

An edge-coloring of a graph G with colors 1 , 2 , … , 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 2 n ) is known only for n ≤ 4 . The second author showed that if n = p 2 q , where p is odd and q is nonnegative, then W ( K 2 n ) ≥ 4 n − 2 − p − q . Later, he conjectured that if n ∈ N , then W ( K 2 n ) = 4 n − 2 − ⌊ log 2 n ⌋ − ‖ n 2 ‖ , where ‖ n 2 ‖ is the number of 1’s in the binary representation of n . In this paper we introduce a new technique to construct interval colorings of complete graphs based on their 1-factorizations, which is used to disprove the conjecture, improve lower and upper bounds on W ( K 2 n ) and determine its exact values for n ≤ 12 .

A Tight Analysis of Kierstead-Trotter Algorithm for Online Unit Interval Coloring

Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for the online interval coloring problem. In this paper, for online unit interval coloring, we show that the number of colors used by the Kierstead-Trotter algorithm is at most $3 \omega(G) - 3$, where $\omega(G)$ is the size of the maximum clique in a given graph $G$, and it is the best possible.

Online and Quasi-online Colorings of Wedges and Intervals

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 $\Theta(\log N/k)$ 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 are enough to color wedges (independent of $N$ and $k$). We also consider quasi-online coloring of intervals. In all cases we present efficient coloring algorithms.

On the interval chromatic number of proper interval graphs

A perfect graph is a graph every subgraph of which has a chromatic number equal to its clique number (Berge, 1963; Lovász, 1972). A (vertex) weighted graph is a graph with a weight function w on its vertices. An interval coloring of a weighted graph maps each vertex v to an interval of size w ( v ) such that the intervals corresponding to adjacent vertices do not intersect. The size of a coloring is the total size of the union of these intervals. The minimum possible size of an interval coloring of a given weighted graph is its interval chromatic number. The clique number of a weighted graph is the maximum weight of a clique in it. Clearly, the interval chromatic number of a weighted graph is at least its clique number. A graph is superperfect if for every weight function on its vertices, the chromatic number of the weighted graph is equal to its clique number (Hoffman, 1974). It is known that determining the interval chromatic number of a given interval graph is Np -Complete , implying that interval graphs are not included in the family of superperfect graphs (Golumbic, 2004). The question whether these results hold for simple graph families is open since then. We answer this question affirmatively for proper interval graphs for which most investigated problems are polynomial ( http://www.graphclasses.org/classes/gc_298.html ). Specifically, we show that determining the interval chromatic number of a proper interval graph is Np -Complete in the strong sense. We present a simple 2-approximation algorithm for this special case, whereas the best known approximation algorithm for interval graphs is a ( 2 + ϵ ) -approximation (Buchsbaum et al., 2004).

On interval Δ-coloring of bipartite graphs

There exists an important class of problems where the minimal-length scheduling comes to regular coloring of a bipartite graph with the least possible number of colors, and the scheduling without downtimes, to the interval coloring of a bipartite graph. Consideration was given to the problem of interval Δ-coloring of the bipartite multigraph. An example was built of the (6, 3)-biregular graph having no interval Δ-coloring.

On Interval Edge Colorings of Biregular Bipartite Graphs With Small Vertex Degrees

AbstractA proper edge coloring of a graph with colors 1, 2, 3, … is called an interval coloring if the colors on the edges incident to each vertex form an interval of integers. A bipartite graph is ‐biregular if every vertex in one part has degree a and every vertex in the other part has degree b. It has been conjectured that all such graphs have interval colorings. We prove that all (3, 6)‐biregular graphs have interval colorings and that all (3, 9)‐biregular graphs having a cubic subgraph covering all vertices of degree 9 admit interval colorings. Moreover, we prove that slightly weaker versions of the conjecture hold for (3, 5)‐biregular, (4, 6)‐biregular, and (4, 8)‐biregular graphs. All our proofs are constructive and yield polynomial time algorithms for constructing the required colorings.

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$ pairwise intersect and $\chi(\mathcal{F})>k$. This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions, we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries, and equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.

Interval Non‐edge‐Colorable Bipartite Graphs and Multigraphs

AbstractAn edge‐coloring of a graph G with colors 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 that has no interval coloring. Erdős's counterexample is the smallest (in a sense of maximum degree) known bipartite graph that 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 article, we give some methods for constructing of interval non‐edge‐colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs that have no interval coloring, contain 20, 19, 21 vertices and have maximum degree 11, 12, 13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.

On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles

A proper edge t-coloring of a graph G is a coloring of its edges with colors 1,2,???,t such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each its vertex x, either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. For an arbitrary simple cycle, all possible values of t are found, for which the graph has a cyclically interval t-coloring.

Scratchpad memory allocation for arrays in permutation graphs

In modern embedded systems, on-chip memory is generally organized as software-managed scratchpad memory (SPM). Li et al. discovered that in many embedded applications, the array interference graphs tend to be superperfect graphs. Based on this, they presented a superperfect graph-based SPM allocation algorithm, which is the best in the literature. In this paper, we make further progress in proving that the array interference graphs they focused on are permutation graphs; that is, a subclass of a superperfect graph. Compared to the latter, the permutation graph has certain advantages, such as linear time recognition and linear time optimal interval coloring. Based on our theoretical results, we improve their algorithm by transforming it into a permutation graph-based one by means of simple plug-in revisions. The experiments confirm that the improved algorithm achieves optimal coloring, and therefore, better allocation than the original superperfect graph-based algorithm for a broader scope of array interference graphs.

A Note on Path Factors of $(3,4)$-Biregular Bipartite Graphs

A proper edge coloring of a graph $G$ with colors $1,2,3,\dots$ is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graph is $(3,4)$-biregular if all vertices in one part have degree $3$ and all vertices in the other part have degree $4$. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph $G$ has a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in $\{2, 4, 6, 8\}$, then $G$ has an interval coloring. It was also conjectured that every simple $(3,4)$-biregular bipartite graph has such a subgraph. We provide some evidence for this conjecture by proving that a simple $(3,4)$-biregular bipartite graph has a spanning subgraph whose components are nontrivial paths with endpoints at $3$-valent vertices and lengths not exceeding $22$.

Max-coloring and online coloring with bandwidths on interval graphs

Given a graph G = ( V, E ) and positive integral vertex weights w : V → N , the max-coloring problem seeks to find a proper vertex coloring of G whose color classes C 1 , C 2 , …, C k , minimize ∑ i =1 k max v ∈ C i w ( v ). This problem, restricted to interval graphs, arises whenever there is a need to design dedicated memory managers that provide better performance than the general-purpose memory management of the operating system. Though this problem seems similar to the dynamic storage allocation problem, there are fundamental differences. We make a connection between max-coloring and online graph coloring and use this to devise a simple 2-approximation algorithm for max-coloring on interval graphs. We also show that a simple first-fit strategy, that is a natural choice for this problem, yields an 8-approximation algorithm. We show this result by proving that the first-fit algorithm for online coloring an interval graph G uses no more than 8 ċ χ( G ) colors, significantly improving the bound of 26 ċ χ( G ) by Kierstead and Qin [1995]. We also show that the max-coloring problem is NP-hard. The problem of online coloring of intervals with bandwidths is a simultaneous generalization of online interval coloring and online bin packing. The input is a set I of intervals, each interval i ∈ I having an associated bandwidth b ( i ) ∈ (0, 1]. We seek an online algorithm that produces a coloring of the intervals such that for any color c and any real r , the sum of the bandwidths of intervals containing r and colored c is at most 1. Motivated by resource allocation problems, Adamy and Erlebach [2003] consider this problem and present an algorithm that uses at most 195 times the number of colors used by an optimal offline algorithm. Using the new analysis of first-fit coloring of interval graphs, we show that the Adamy-Erlebach algorithm is 35-competitive. Finally, we generalize the Adamy-Erlebach algorithm to a class of algorithms and show that a different instance from this class is 30-competitive.

Interval coloring of (3, 4)-biregular bigraphs having two [formula omitted]-biregular bipartite subgraphs

An interval coloring of a graph is a proper edge coloring such that the set of colors used at every vertex is an interval of integers. An ( α , β )-biregular bigraph is a bipartite graph in which each vertex in one part has degree α and each vertex in the other part has degree β . It is unknown whether all ( 3 , 4 ) -biregular bigraphs have interval colorings. In this work we prove that if a ( 3 , 4 ) -biregular bigraph G = ( X , Y ; E ) has two (2, 3)-biregular bipartite subgraphs G 1 = ( Y , X 1 ; E 1 ) , G 2 = ( Y , X 2 ; E 2 ) such that X 1 ∪ X 2 = X , E 1 ∪ E 2 = E , X 1 ∩ X 2 = 0̸ , and E 1 ∩ E 2 = 0̸ , then G has an interval 6-coloring.

On resistance of graphs

An edge-coloring of a graph G with integers is called an interval 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. It is known that not all graphs have interval colorings, and therefore it is expedient to consider a measure of closeness for a graph to be interval colorable. In this paper we introduce such a measure (resistance of a graph) and we determine the exact value of the resistance for some classes of graphs.

Scratchpad memory allocation for data aggregates via interval coloring in superperfect graphs

Existing methods place data or code in scratchpad memory (SPM) by relying on heuristics or resorting to integer programming or mapping it to a graph-coloring problem. In this article, the SPM allocation problem for arrays is formulated as an interval coloring problem. The key observation is that in many embedded C programs, two arrays can be modeled such that either their live ranges do not interfere or one contains the other (with good accuracy). As a result, array interference graphs often form a special class of superperfect graphs (known as comparability graphs), and their optimal interval colorings become efficiently solvable. This insight has led to the development of an SPM allocation algorithm that places arrays in an interference graph in SPM by examining its maximal cliques. If the SPM is no smaller than the clique number of an interference graph, then all arrays in the graph can be placed in SPM optimally. Otherwise, we rely on containment-motivated heuristics to split or spill array live ranges until the resulting graph is optimally colorable. We have implemented our algorithm in SUIF/machSUIF and evaluated it using a set of embedded C benchmarks from MediaBench and MiBench. Compared to a graph-coloring algorithm and an optimal ILP algorithm (when it runs to completion), our algorithm achieves close-to-optimal results and is superior to graph coloring for the benchmarks tested.

On a reduction of the interval coloring problem to a series of bandwidth coloring problems

Given a graph G=(V,E) with strictly positive integer weights ? i on the vertices i?V, an interval coloring of G is a function I that assigns an interval I(i) of ? i consecutive integers (called colors) to each vertex i?V so that I(i)?I(j)=? for all edges {i,j}?E. The interval coloring problem is to determine an interval coloring that uses as few colors as possible. Assuming that a strictly positive integer weight ? ij is associated with each edge {i,j}?E, a bandwidth coloring of G is a function c that assigns an integer (called a color) to each vertex i?V so that |c(i)?c(j)|?? ij for all edges {i,j}?E. The bandwidth coloring problem is to determine a bandwidth coloring with minimum difference between the largest and the smallest colors used. We prove that an optimal solution of the interval coloring problem can be obtained by solving a series of bandwidth coloring problems. Computational experiments demonstrate that such a reduction can help to solve larger instances or to obtain better upper bounds on the optimal solution value of the interval coloring problem.

About equivalent interval colorings of weighted graphs

Given a graph G = ( V , E ) with strictly positive integer weights ω i on the vertices i ∈ V , a k -interval coloring of G is a function I that assigns an interval I ( i ) ⊆ { 1 , … , k } of ω i consecutive integers (called colors) to each vertex i ∈ V . If two adjacent vertices x and y have common colors, i.e. I ( i ) ∩ I ( j ) ≠ 0̸ for an edge [ i , j ] in G , then the edge [ i , j ] is said conflicting . A k -interval coloring without conflicting edges is said legal . The interval coloring problem (ICP) is to determine the smallest integer k , called interval chromatic number of G and denoted χ i n t ( G ) , such that there exists a legal k -interval coloring of G . For a fixed integer k , the k -interval graph coloring problem ( k -ICP) is to determine a k -interval coloring of G with a minimum number of conflicting edges. The ICP and k -ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω i = 1 for all vertices i ∈ V ). Two k -interval colorings I 1 and I 2 are said equivalent if there is a permutation π of the integers 1 , … , k such that ℓ ∈ I 1 ( i ) if and only if π ( ℓ ) ∈ I 2 ( i ) for all vertices i ∈ V . As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k -ICP can be increased by avoiding considering equivalent k -interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k -interval colorings. We then show how a simple tabu search algorithm for the k -ICP can possibly be improved by forbidding the visit of equivalent solutions.

Proper path‐factors and interval edge‐coloring of (3,4)‐biregular bigraphs

AbstractAn 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 several sufficient conditions for the existence of such a subgraph. © 2009 Wiley Periodicals, Inc. J Graph Theory

Variable Sized Online Interval Coloring with Bandwidth

We consider online coloring of intervals with bandwidth in a setting where colors have variable capacities. Whenever the algorithm opens a new color, it must choose the capacity for that color and cannot change it later. A set of intervals can be assigned the same color a of capacity C a if the sum of bandwidths of intervals at each point does not exceed C a . The goal is to minimize the total capacity of all the colors used. We consider the bounded model, where all capacities must be chosen in the range (0,1], and the unbounded model, where the algorithm may use colors of any positive capacity. For the absolute competitive ratio, we give an upper bound of 14 and a lower bound of 4.59 for the bounded model, and an upper bound of 4 and a matching lower bound of 4 for the unbounded model. We also consider the offline version of these problems and show that whereas the unbounded model is polynomially solvable, the bounded model is NP-hard in the strong sense and admits a 3.6-approximation algorithm.