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Added by
Joan Boyar
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In graph coloring problems, the goal is to assign a positive integer color to each vertex of an input graph such that adjacent vertices do not receive the same color assignment. For classic graph coloring, the goal is to minimize the maximum color used, and for the sum coloring problem, the goal is to minimize the sum of colors assigned to all input vertices. In the offline variant, the entire graph is presented at once, and in online problems, one vertex is presented for coloring at each time, and the only information is the identity of its neighbors among previously known vertices. In batched graph coloring, vertices are presented in k batches, for a fixed integer k > 1, such that the vertices of a batch are presented as a set, and must be colored before the vertices of the next batch are presented. This last model is an intermediate model, which bridges between the two extreme scenarios of the online and offline models. We provide several results, including a general result for sum coloring and results for the classic graph coloring problem on restricted graph classes: We show tight bounds for any graph class containing trees as a subclass (e.g., forests, bipartite graphs, planar graphs, and perfect graphs), and a surprising result for interval graphs and k = 2, where the value of the (strict and asymptotic) competitive ratio depends on whether the graph is presented with its interval representation or not.
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In this work, we continue the study of vertex colorings of graphs, in which adjacent vertices are allowed to be of the same color as long as each monochromatic connected component is of relatively small cardinality. We focus on colorings with two and three available colors and present improved bounds on the size of the monochromatic connected components for two meaningful subclasses of planar graphs, namely maximal outerplanar graphs and complete planar 3-trees.
Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by $k$: (1) Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal results in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for every $vin V(G)$, decide whether $G$ has a proper list coloring; (2) Given a graph $G$, a clique modulator $D$ of size $k$ for $G$, and a pre-coloring $lambda_P: X rightarrow Q$ for $X subseteq V(G),$ decide whether $lambda_P$ can be extended to a proper coloring of $G$ using only colors from $Q.$ For Problem 1 we design an $O^*(2^k)$-time randomized algorithm and for Problem 2 we obtain a kernel with at most $3k$ vertices. Banik et al. (IWOCA 2019) proved the the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph $G$, an integer $k$, and a list $L(v)$ of exactly $n-k$ colors for every $v in V(G),$ decide whether there is a proper list coloring for $G.$ We obtain a kernel with $O(k^2)$ vertices and colors and a compression to a variation of the problem with $O(k)$ vertices and $O(k^2)$ colors.
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 $O(mathcal{C} d N^{1/d})$-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 $O(d N^{1/d})$ 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 $Omega(N^frac{2}{c(c-1)})$ vertices per update, for any constant $c geq 2$.
For a given graph $G$, the least integer $kgeq 2$ such that for every Abelian group $mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)rightarrow mathcal{G}$ so that $sum_{xin N(u)}f(xu) eq sum_{xin N(v)}f(xv)$ for each edge $uvin E(G)$ is called the textit{group twin chromatic index} of $G$ and denoted by $chi_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $chi_g(G)leq Delta(G)+3$ for all graphs without isolated edges, where $Delta(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $chi_g(G)leq 2(Delta(G)+{rm col}(G))-5$, where ${rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $mathbb{Z}_k$.
We present a randomized distributed algorithm that computes a $Delta$-coloring in any non-complete graph with maximum degree $Delta geq 4$ in $O(log Delta) + 2^{O(sqrt{loglog n})}$ rounds, as well as a randomized algorithm that computes a $Delta$-coloring in $O((log log n)^2)$ rounds when $Delta in [3, O(1)]$. Both these algorithms improve on an $O(log^3 n/log Delta)$-round algorithm of Panconesi and Srinivasan~[STOC1993], which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an $Omega(loglog n)$ round lower bound of Brandt et al.~[STOC16].
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