Stable Kneser Graphs are almost all not weakly Hom-Idempotent

A graph G is said to be hom-idempotent if there is an homomorphism from G 2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from G n+1 to G n . Larose et al. [Eur. J. Comb. 19:867-881, 1998] proved that Kneser graphs KG(n, k) are not weakly hom-idempotent for n ≥ 2k + 1, k ≥ 2. We show that 2-stable Kneser graphs KG(n, k) 2−stab are not weakly hom-idempotent, for n ≥ 2k + 2, k ≥ 2. Moreover, for s, k ≥ 2, we prove that s-stable Kneser graphs KG(ks+1, k) s−stab are circulant graphs and so hom-idempotent graphs

A distributed approximation algorithm for the minimum degree minimum weight spanning trees

International audienceFischer has shown how to compute a minimum weight spanning tree of degree at most $b \Delta^* + \lceil \log_b n\rceil$ in time $O(n^{4 + 1/\!\ln b})$ for any constant $b > 1$, where $\Delta^*$ is the value of an optimal solution and $n$ is the number of nodes in the network. In this paper, we propose a distributed version of Fischer's algorithm that requires messages and time complexity $O(n^{2 + 1/\!\ln b})$, and $O(n)$ space per node

k-tuple colorings of the Cartesian product of graphs

A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G such that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G, χk(G), is the smallest t so that there is a k-tuple coloring of G using t colors. It is well known that χ(G□H)=max{χ(G),χ(H)}. In this paper, we show that there exist graphs G and H such that χk(G□H)>max{χk(G),χk(H)} for k≥2. Moreover, we also show that there exist graph families such that, for any k≥1, the k-tuple chromatic number of their Cartesian product is equal to the maximum k-tuple chromatic number of its factors.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Koch, Ivo Valerio. Universidad Nacional de General Sarmiento. Instituto de Industria; ArgentinaFil: Torres, Pablo. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Valencia Pabon, Mario. Universite de Paris 13-Nord; Francia. Centre National de la Recherche Scientifique; Franci

On the diameter of Schrijver graphs

For k ≥ 1 and n ≥ 2k, the well known Kneser graph KG(n, k) has all k-element subsets of an n-element set as vertices; two such subsets are adjacent if they are disjoint. Schrijver constructed a vertex-critical subgraph SG(n, k) of KG(n, k) with the same chromatic number. In this paper, we compute the diameter of the graph SG(2k + r,k) with r ≥ 1. We obtain that the diameter of SG(2k + r, k) is equal to 2 if r ≥ 2k - 2; 3 if k≥ - 2 ≤ r ≤ 2k - 3; k if r = 1; and for 2 ≤ r ≤ k - 3, we obtain that the diameter of SG(2k + r, k) is at most equal to k - r + 1.Fil: Pastine, Adrián Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; ArgentinaFil: Torres, Pablo Daniel. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe; ArgentinaFil: Valencia Pabon, Mario. Universite Sorbonne Paris Nord; FranciaXI Latin and American Algorithms, Graphs and Optimization Symposium.Sao PauloBrasilUniversity of Sao Paul

The permutation-path coloring problem on trees

AbstractIn this paper we first show that the permutation-path coloring problem is NP-hard even for very restrictive instances like involutions, which are permutations that contain only cycles of length at most two, on both binary trees and on trees having only two vertices with degree greater than two, and for circular permutations, which are permutations that contain exactly one cycle, on trees with maximum degree greater than or equal to 4. We calculate a lower bound on the average complexity of the permutation-path coloring problem on arbitrary networks. Then we give combinatorial and asymptotic results for the permutation-path coloring problem on linear networks in order to show that the average number of colors needed to color any permutation on a linear network on n vertices is n/4+o(n). We extend these results and obtain an upper bound on the average complexity of the permutation-path coloring problem on arbitrary trees, obtaining exact results in the case of generalized star trees. Finally we explain how to extend these results for the involutions-path coloring problem on arbitrary trees

A one-to-one correspondence between potential solutions of the cluster deletion problem and the minimum sum coloring problem, and its application to P 4 -sparse graphs

In this note we show a one-to-one correspondence between potentially optimal solutions to the cluster deletion problem in a graph G and potentially optimal solutions for the minimum sum coloring problem in G (i.e. the complement graph of G). We apply this correspondence to polynomially solve the cluster deletion problem in a subclass of P 4 -sparse graphs that strictly includes P 4 -reducible graphs

Computing and Counting Longest Paths on Circular-Arc Graphs in Polynomial Time

The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O(n4)) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph G′ of G admits a path P′ on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O(n4) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight

Minimum Sum Edge Colorings of Multicycles

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength} of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems