Optimality Clue for Graph Coloring Problem

In this paper, we present a new approach which qualifies or not a solution found by a heuristic as a potential optimal solution. Our approach is based on the following observation: for a minimization problem, the number of admissible solutions decreases with the value of the objective function. For the Graph Coloring Problem (GCP), we confirm this observation and present a new way to prove optimality. This proof is based on the counting of the number of different k-colorings and the number of independent sets of a given graph G. Exact solutions counting problems are difficult problems (\#P-complete). However, we show that, using only randomized heuristics, it is possible to define an estimation of the upper bound of the number of k-colorings. This estimate has been calibrated on a large benchmark of graph instances for which the exact number of optimal k-colorings is known. Our approach, called optimality clue, build a sample of k-colorings of a given graph by running many times one randomized heuristic on the same graph instance. We use the evolutionary algorithm HEAD [Moalic et Gondran, 2018], which is one of the most efficient heuristic for GCP. Optimality clue matches with the standard definition of optimality on a wide number of instances of DIMACS and RBCII benchmarks where the optimality is known. Then, we show the clue of optimality for another set of graph instances. Optimality Metaheuristics Near-optimal

Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult

We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem

Extremal H-Colorings of Graphs with Fixed Minimum Degree

For graphs G and H, a homomorphism from G to H, or H-coloring of G, is a map from the vertices of G to the vertices of H that preserves adjacency. When H is composed of an edge with one looped endvertex, an H-coloring of G corresponds to an independent set in G. Galvin showed that, for sufficiently large n, the complete bipartite graph Kδ,n-δ is the n-vertex graph with minimum degree δ that has the largest number of independent sets. In this paper, we begin the project of generalizing this result to arbitrary H. Writing hom(G, H) for the number of H-colorings of G, we show that for fixed H and δ = 1 or δ = 2, hom(G, H) ≤ max{hom(Kδ+1,H)n⁄δ =1, hom(Kδ,δ,H)n⁄2δ, hom(Kδ,n-δ,H)} for any n-vertex G with minimum degree δ (for sufficiently large n). We also provide examples of H for which the maximum is achieved by hom(Kδ+1, H)n⁄δ+1 and other H for which the maximum is achieved by hom(Kδ,δ,H)n⁄2δ. For δ ≥ 3 (and sufficiently large n), we provide a infinite family of H for which hom(G, H) ≤ hom (Kδ,n-δ, H) for any n-vertex G with minimum degree δ. The results generalize to weighted H-colorings

Questions on the Structure of Perfect Matchings inspired by Quantum Physics

We state a number of related questions on the structure of perfect matchings. Those questions are inspired by and directly connected to Quantum Physics. In particular, they concern the constructability of general quantum states using modern photonic technology. For that we introduce a new concept, denoted as inherited vertex coloring. It is a vertex coloring for every perfect matching. The colors are inherited from the color of the incident edge for each perfect matching. First, we formulate the concepts and questions in pure graph-theoretical language, and finally we explain the physical context of every mathematical object that we use. Importantly, every progress towards answering these questions can directly be translated into new understanding in quantum physics.Comment: 10 pages, 4 figures, 6 questions (added suggestions from peer-review