44 research outputs found
A characterization of b-chromatic and partial Grundy numbers by induced subgraphs
Gy{\'a}rf{\'a}s et al. and Zaker have proven that the Grundy number of a
graph satisfies if and only if contains an induced
subgraph called a -atom.The family of -atoms has bounded order and
contains a finite number of graphs.In this article, we introduce equivalents of
-atoms for b-coloring and partial Grundy coloring.This concept is used to
prove that determining if and (under
conditions for the b-coloring), for a graph , is in XP with parameter .We
illustrate the utility of the concept of -atoms by giving results on
b-critical vertices and edges, on b-perfect graphs and on graphs of girth at
least
The b-chromatic number of power graphs
The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles
Incidence, a Scoring Positional Game on Graphs
Positional games have been introduced by Hales and Jewett in 1963 and have
been extensively investigated in the literature since then. These games are
played on a hypergraph where two players alternately select an unclaimed vertex
of it. In the Maker-Breaker convention, if Maker manages to fully take a
hyperedge, she wins, otherwise, Breaker is the winner. In the Maker-Maker
convention, the first player to take a hyperedge wins. In both cases, the game
stops as soon as Maker has taken a hyperedge. By definition, this family of
games does not handle scores and cannot represent games in which players want
to maximize a quantity.
In this work, we introduce scoring positional games, that consist in playing
on a hypergraph until all the vertices are claimed, and by defining the score
as the number of hyperedges a player has fully taken. We focus here on
Incidence, a scoring positional game played on a 2-uniform hypergraph, i.e. an
undirected graph. In this game, two players alternately claim the vertices of a
graph and score the number of edges for which they own both end vertices. In
the Maker-Breaker version, Maker aims at maximizing the number of edges she
owns, while Breaker aims at minimizing it. In the Maker-Maker version, both
players try to take more edges than their opponent.
We first give some general results on scoring positional games such that
their membership in Milnor's universe and some general bounds on the score. We
prove that, surprisingly, computing the score in the Maker-Breaker version of
Incidence is PSPACE-complete whereas in the Maker-Maker convention, the
relative score can be obtained in polynomial time. In addition, for the
Maker-Breaker convention, we give a formula for the score on paths by using
some equivalences due to Milnor's universe. This result implies that the score
on cycles can also be computed in polynomial time
A note on Grundy colorings of central graphs
International audienceA Grundy coloring of a graph G is a proper vertex coloring of G where any vertex x, colored with c(x), has a neighbor of any color 1,2,...,c(x)-1. A central graph G^c is obtained from G by adding an edge between any two non adjacent vertices in G and subdividing any edge of G once. In this note we focus on Grundy colorings of central graphs. We present some bounds related to parameters of G and a Nordhaus-Gaddum inequality. We also determine exact values for the Grundy coloring of some central classical graphs
The b_q-coloring of graphs
International audienceIn this article we introduce a new graph coloring called the b_q-coloring. A b_q-coloring of a graph G is a proper vertex coloring of G with k colors such that every color class c admits a set of vertices S, of size at most q, such that every color except c appears in the neighborhood of S. The aim of this coloring is to generalize the domination constraint given in the b-coloring of a graph where every color admits only one dominating vertex (adjacent to every other color). The largest positive integer k for which a graph has a b_q-coloring using k colors is the b_q-chromatic number. We present some classes of graphs for which the b_q-chromatic number has maximum value and we give exact values of this parameter for paths and cycles. We also present some bounds for Cartesian products of graphs
Generation of Valid Labeled Binary Trees
International audienceGenerating binary trees is a well-known problem. In this paper, we add some constraints to leaves of these trees. Such trees are used in the morphing of polygons, where a polygon P is represented by a binary tree T and each angle of P is a weight on a leaf of T. In the following, we give two algorithms to generate all binary trees, without repetitions, having the same weight distribution to their leaves and representing all parallel polygons to P
The b-chromatic number of power graphs of complete caterpillars
International audienceLet G be a graph on vertices v1, v2, . . . , vn. The b-chromatic number of G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and every color i admits a representant x adjacent to a vertex of each color j, 1 ≤ j i ≤ k. In this paper, we give the exact value for the b-chromatic number of power graphs of a complete caterpillar
Generation of Unordered Binary Trees
International audienceA binary unordered tree is a tree where each internal node has two children and the relative order of the subtrees of a node is not important (i.e. two trees are different if they differ only in the respective ordering of subtrees of nodes). We present a new method to generate all binary rooted unordered trees with n internal nodes, without duplications, in O(log n) time
A note on Grundy colorings of central graphs
International audienceA Grundy coloring of a graph G is a proper vertex coloring of G where any vertex x, colored with c(x), has a neighbor of any color 1,2,...,c(x)-1. A central graph G^c is obtained from G by adding an edge between any two non adjacent vertices in G and subdividing any edge of G once. In this note we focus on Grundy colorings of central graphs. We present some bounds related to parameters of G and a Nordhaus-Gaddum inequality. We also determine exact values for the Grundy coloring of some central classical graphs
The b-chromatic number of power graphs
International audienceThe b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles