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Improved EdgeColoring Algorithms for Planar Graphs
 JOURNAL OF ALGORITHMS
, 1996
"... We consider the problem of edgecoloring planar graphs. It is known that a planar graph G with maximum degree \Delta \geq 8 can be colored with \Delta colors. We present two algorithms which find such a coloring when \Delta \geq 9. The first one is a sequential O(n log n) time algorithm. The other o ..."
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We consider the problem of edgecoloring planar graphs. It is known that a planar graph G with maximum degree \Delta \geq 8 can be colored with \Delta colors. We present two algorithms which find such a coloring when \Delta \geq 9. The first one is a sequential O(n log n) time algorithm. The other one is a parallel EREW PRAM algorithm which works in time O(log³ n) and uses O(n) processors.
Information Multicast in (Pseudo)Planar Networks: Efficient Network Coding over Small Finite Fields
"... Abstract—Network coding encourages innetwork mixing of information flows for enhanced network capacity, particularly for multicast data dissemination. This work aims to explore properties in the underlying network topology for efficient network coding solutions, including efficient code assignment ..."
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Abstract—Network coding encourages innetwork mixing of information flows for enhanced network capacity, particularly for multicast data dissemination. This work aims to explore properties in the underlying network topology for efficient network coding solutions, including efficient code assignment algorithms and efficient encoding/decoding operations that come with small base field sizes. The following cases of (pseudo)planar types of networks are studied: outerplanar networks where all nodes colocate on a common face, relay/terminal coface networks where all relay/terminal nodes colocate on a common face, general planar networks, and apex networks. I.
The dprecoloring problem for kdegenerate graphs �
, 2006
"... In this paper we deal with the dPRECOLORING EXTENSION (dPREXT) problem in various classes of graphs. The dPREXT problem is the special case of PRECOLORING EXTENSION problem where, for a fixed constant d, input instances are restricted to contain at most d precolored vertices for every available c ..."
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In this paper we deal with the dPRECOLORING EXTENSION (dPREXT) problem in various classes of graphs. The dPREXT problem is the special case of PRECOLORING EXTENSION problem where, for a fixed constant d, input instances are restricted to contain at most d precolored vertices for every available color. The goal is to decide if there exists an extension of given precoloring using only available colors or to find it. We present a linear time algorithm for both, the decision and the search version of dPREXT, in the following cases: (i) restricted to the class of kdegenerate graphs (hence also planar graphs) and with sufficiently large set S of available colors, and (ii) restricted to the class of partial ktrees (without any size restriction on S). We also study the following problem related to dPREXT: given an instance of the dPREXT problem which is extendable by colors of S, what is the minimum number of colors of S sufficient to use for precolorless vertices over all such extensions? We establish lower and upper bounds on this value for kdegenerate graphs and its various subclasses (e.g., planar graphs, outerplanar graphs) and prove tight results for the class of trees.