9,479 research outputs found
Channel Assignment with Separation on Trees and Interval Graphs
Given a vector of non increasing positive integers, and an undirected graph , an -coloring of is a function from the vertex set to a set of nonnegative integers such that , if where is the distance (i.e. the minimum number of edges) between the vertices and . An optimal -coloring for is one minimizing the largest used integer over all such colorings. This coloring problem has relevant application in channel assignment for interference avoidance in wireless networks, where channels (i.e. colors) assigned to interfering stations (i.e. vertices) at distance must be at least apart, while the same channel can be reused only at stations whose distance is larger than . This paper presents efficient algorithms for finding optimal -colorings of trees and interval graphs as well as optimal -colorings of complete binary trees. Moreover, efficient algorithms are also provided for finding approximate -colorings of trees and interval graphs as well as approximate -colorings of unit interval graphs
Solution of Vizing's Problem on Interchanges for Graphs with Maximum Degree 4 and Related Results
Let be a Class 1 graph with maximum degree and let be an
integer. We show that any proper -edge coloring of can be transformed to
any proper -edge coloring of using only transformations on -colored
subgraphs (so-called interchanges). This settles the smallest previously
unsolved case of a well-known problem of Vizing on interchanges, posed in 1965.
Using our result we give an affirmative answer to a question of Mohar for two
classes of graphs: we show that all proper -edge colorings of a Class 1
graph with maximum degree 4 are Kempe equivalent, that is, can be transformed
to each other by interchanges, and that all proper 7-edge colorings of a Class
2 graph with maximum degree 5 are Kempe equivalent
On the Total Set Chromatic Number of Graphs
Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χs(G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings, that is, we study set colorings of the total graph of graphs. Given a graph G = (V; E); its total graph T (G) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First; we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs
On the Total Set Chromatic Number of Graphs
Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χs(G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = (V, E), its total graph T(G) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First, we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs
b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs
A b-coloring of a graph is a proper coloring such that every color class
contains a vertex that is adjacent to all other color classes. The b-chromatic
number of a graph G, denoted by \chi_b(G), is the maximum number t such that G
admits a b-coloring with t colors. A graph G is called b-continuous if it
admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and
b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of
G, and every induced subgraph H_2 of H_1.
We investigate the b-chromatic number of graphs with stability number two.
These are exactly the complements of triangle-free graphs, thus including all
complements of bipartite graphs. The main results of this work are the
following:
- We characterize the b-colorings of a graph with stability number two in
terms of matchings with no augmenting paths of length one or three. We derive
that graphs with stability number two are b-continuous and b-monotonic.
- We prove that it is NP-complete to decide whether the b-chromatic number of
co-bipartite graph is at most a given threshold.
- We describe a polynomial time dynamic programming algorithm to compute the
b-chromatic number of co-trees.
- Extending several previous results, we show that there is a polynomial time
dynamic programming algorithm for computing the b-chromatic number of
tree-cographs. Moreover, we show that tree-cographs are b-continuous and
b-monotonic
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