Dynamic Graph Coloring Research Articles

An improved upper bound for the dynamic list coloring of 1-planar graphs

<abstract><p>A graph is $ 1 $-planar if it can be drawn in the plane such that each of its edges is crossed at most once. A dynamic coloring of a graph $ G $ is a proper vertex coloring such that for each vertex of degree at least 2, its neighbors receive at least two different colors. The list dynamic chromatic number $ ch_{d}(G) $ of $ G $ is the least number $ k $ such that for any assignment of $ k $-element lists to the vertices of $ G $, there is a dynamic coloring of $ G $ where the color on each vertex is chosen from its list. In this paper, we show that if $ G $ is a 1-planar graph, then $ ch_{d}(G)\leq 10 $. This improves a result by Zhang and Li <sup>[<xref ref-type="bibr" rid="b16">16</xref>]</sup>, which says that every 1-planar graph $ G $ has $ ch_{d}(G)\leq 11 $.</p></abstract>

Time Complexity Analysis of Randomized Search Heuristics for the Dynamic Graph Coloring Problem

We contribute to the theoretical understanding of randomized search heuristics for dynamic problems. We consider the classical vertex coloring problem on graphs and investigate the dynamic setting where edges are added to the current graph. We then analyze the expected time for randomized search heuristics to recompute high quality solutions. The (1+1) Evolutionary Algorithm and RLS operate in a setting where the number of colors is bounded and we are minimizing the number of conflicts. Iterated local search algorithms use an unbounded color palette and aim to use the smallest colors and, consequently, the smallest number of colors. We identify classes of bipartite graphs where reoptimization is as hard as or even harder than optimization from scratch, i.e., starting with a random initialization. Even adding a single edge can lead to hard symmetry problems. However, graph classes that are hard for one algorithm turn out to be easy for others. In most cases our bounds show that reoptimization is faster than optimizing from scratch. We further show that tailoring mutation operators to parts of the graph where changes have occurred can significantly reduce the expected reoptimization time. In most settings the expected reoptimization time for such tailored algorithms is linear in the number of added edges. However, tailored algorithms cannot prevent exponential times in settings where the original algorithm is inefficient.

Improved Dynamic Graph Coloring

This article studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within n1-εfor any ε > 0, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring or alternatively, study restricted families of graphs.Toward understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm forC-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. Optimizing the number of recolorings, sometimes referred to as the recourse bound, is important for various practical applications. In WADS ’17, Barba et al. devised two complementary algorithms: for any β > 0, the first (respectively, second) maintains anO(Cβn1/β)(respectively,O(Cβ)-coloring while recoloring O(β) (respectively, O(βn1/β)) vertices per update. Barba et al. also showed that the second trade-off appears to exhibit the right behavior, at least for β = O(1): any algorithm that maintains aC-coloring of ann-vertex dynamic forest must recolor Ω (n2C(C-1)) vertices per update, for any constant C ≥ 2. Our contribution is twofold:• We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: for any β > 0, we get a Ô (Cβlog2n)-coloring with O(β) recolorings per update, where the Ô notation suppressespolyloglog(n)factors. In particular, for β = O(1), we get constant recolorings withpolylog(n)colors; not only is this an exponential improvement over the previous bound but also it unveils a rather surprising phenomenon: the trade-off between the number of colors and recolorings is highly non-symmetric.• For uniformly sparse graphs, we use low out-degree orientations to strengthen the preceding result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest. From this data structure, we get a deterministic algorithm for graphs of arboricity ɑ that maintains an O(ɑ log2n)-coloring in amortized O(1) time.

List [formula omitted]-dynamic coloring of graphs with small maximum average degree

An r-dynamick-coloring of a graphG is a proper k-coloring such that every vertex v in V(G) has neighbors in at least min{d(v),r} different classes. The r-dynamic chromatic number ofG, written χr(G), is the minimum k such that G has such a coloring. The list r-dynamic chromatic number of G is denoted chr(G). In this paper, we prove that every graph G with one of the following conditions has a list (r+1)-dynamic coloring: (1) mad(G)<187 and r≥8; (2) mad(G)<52 and r≥6; (3) mad(G)<125 and r≥5; (4) mad(G)<2.8−ϵ and r≥f(ϵ)=165ϵ−2, where 0<ϵ≤110. Also we prove that every graph G with mad(G)<4−ϵ and r≥f(ϵ)=5M−11 has a list (r+f(ϵ))-dynamic coloring, where 0<ϵ<1, M=⌈8ϵ⌉−2.

On dynamic coloring of certain cycle-related graphs

Coloring the vertices of a particular graph has often been motivated by its utility to various applied fields and its mathematical interest. A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. A dynamic k-coloring of a graph is a dynamic coloring with k colors. A dynamic k-coloring is also called a conditional (k, 2)-coloring. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number chi _d(G) of G. In this paper, we investigate the dynamic chromatic number for the line graph of sunlet graph and middle graph, total graph and central graph of sunlet graphs, paths and cycles. Also, we find the dynamic chromatic number for Mycielskian of paths and cycles and the join graph of paths and cycles.

Dynamic Graph Coloring

In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any d>0, the first algorithm maintains a proper O(mathcal {C} dN ^{1/d})-coloring while recoloring at most O(d) vertices per update, where mathcal {C} and N are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an O(mathcal {C} d)-coloring with O(dN ^{1/d}) recolorings per update. The two converge when d = log N , maintaining an O(mathcal {C} log N)-coloring with O(log N) recolorings per update. We also present a lower bound, showing that any algorithm that maintains a c-coloring of a 2-colorable graph on N vertices must recolor at least varOmega (N ^frac{2}{c(c-1)}) vertices per update, for any constant c ge 2.

Dynamic coloring of graphs having no [formula omitted] minor

We prove that every simple connected graph with no K5 minor admits a proper 4-coloring such that the neighborhood of each vertex v having more than one neighbor is not monochromatic, unless the graph is isomorphic to the cycle of length 5. This generalizes the result on planar graphs by S.-J. Kim, W.-J. Park and the second author [Discrete Appl. Math. 161 (2013) 2207–2212].

Element deletion changes in dynamic coloring of graphs

A proper vertex k-coloring of a graph G is dynamic if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic k-coloring is the dynamic chromatic number χd(G). In this paper the differences between χd(G) and χd(G−e), and between χd(G) and χd(G−v) are investigated respectively.

Dynamic coloring and list dynamic coloring of planar graphs

A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. The dynamic chromatic numberχd(G) of a graph G is the least number k such that G has a dynamic coloring with k colors. We show that χd(G)≤4 for every planar graph except C5, which was conjectured in Chen et al. (2012) [5].The list dynamic chromatic numberchd(G) of G is the least number k such that for any assignment of k-element lists to the vertices of G, there is a dynamic coloring of G where the color on each vertex is chosen from its list. Based on Thomassen’s (1994) result [14] that every planar graph is 5-choosable, an interesting question is whether the list dynamic chromatic number of every planar graph is at most 5 or not. We answer this question by showing that chd(G)≤5 for every planar graph.

Dynamic chromatic number of regular graphs

A k-dynamic coloring of a graph G is a proper coloring of G with k colors such that for every vertex v∈V(G) of degree at least 2, the neighbors of v receive at least 2 colors. The dynamic chromatic number of a graph G, χ2(G), is the least number k such that G admits a k-dynamic coloring. It was conjectured [B. Montgomery, Dynamic coloring of graphs, Ph.D. Thesis, West Virginia University, 2001.] that if G is a k-regular graph, then χ2(G)−χ(G)≤2. In this paper, we prove that if G is a k-regular graph with χ(G)≥4, then χ2(G)≤χ(G)+α(G2). It confirms the conjecture for all regular graphs with diameter at most 2 and chromatic number at least 4. In fact, it shows that χ2(G)−χ(G)≤1 provided that G has diameter at most 2 and χ(G)≥4. Moreover, we show that for any k-regular graph G with no induced C4, χ2(G)−χ(G)≤2⌈4lnk+1⌉. Also, we show that for any n there exists a regular graph G whose chromatic number is n and χ2(G)−χ(G)≥1. This result gives a negative answer to a conjecture of Ahadi et al. [A. Ahadi, S. Akbari, A. Dehghan, M. Ghanbari, On the difference between chromatic number and dynamic chromatic number of graphs, Discrete Math. Available online 2011].

Dynamic Coloring of Graphs

Dynamics is an inherent feature of many real life systems so it is natural to define and investigate the properties of models that reflect their dynamic nature. Dynamic graph colorings can be naturally applied in system modeling, e.g. for scheduling threads of parallel programs, time sharing in wireless networks, session scheduling in high-speed LAN's, channel assignment in WDM optical networks as well as traffic scheduling. In the dynamic setting of the problem, a graph we color is not given in advance and new vertices together with adjacent edges are revealed one after another at algorithm's input during the coloring process. Moreover, independently of the algorithm, some vertices may lose their colors and the algorithm may be asked to color them again. We formally define a dynamic graph coloring problem, the dynamic chromatic number and prove various bounds on its value. We also analyze the effectiveness of the dynamic coloring algorithm Dynamic-Fit for selected classes of graphs. In particular, we deal with trees, products of graphs and classes of graphs for which Dynamic-Fit is competitive. Motivated by applications, we state the problem of dynamic coloring with discoloring constraints for which the performance of the dynamic algorithm Time-Fit is analyzed and give a characterization of graphs k-critical for Time-Fit. Since for any fixed k > 0 the number of such graphs is finite, it is possible to decide in polynomial time whether Time-Fit will always color a given graph with at most k colors.

On the dynamic coloring of graphs

A dynamic coloring of a graph G is a proper coloring such that, for every vertex v ∈ 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, for every k -regular graph G , χ 2 ( G ) ≤ χ ( G ) + 14.06 ln k + 1 . Also, we introduce an upper bound for the dynamic list chromatic number of regular graphs.

On the list dynamic coloring of graphs

A proper vertex coloring of a graph G is called a dynamic coloring if for every vertex v of degree at least 2, the neighbors of v receive at least two different colors. Assume that ch 2 ( G ) is the minimum number k such that for every list assignment of size k to each vertex of G , there is a dynamic coloring of G such that every vertex is colored with a color from its list. In this paper, it is proved that if G is a graph with no component isomorphic to C 5 and Δ ( G ) ≥ 3 , then ch 2 ( G ) ≤ Δ ( G ) + 1 , where Δ ( G ) is the maximum degree of G . This generalizes a result due to Lai, Montgomery and Poon which says that under the same assumptions χ 2 ( G ) ≤ Δ ( G ) + 1 . Among other results, we determine ch 2 ( C n ) , for every natural number n .