Positive graphs

We study “positive” graphs that have a nonnegative homomorphism number into every edge-weighted graph (where the edgeweights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary simple graph and gluing them together along an independent set of nodes. We prove the conjecture for various classes of graphs including all trees. We prove a number of properties of positive graphs, including the fact that they have a homomorphic image which has at least half the original number of nodes but in which every edge has an even number of pre-images. The results, combined with a computer program, imply that the conjecture is true for all graphs up to 9 nodes. Contents 1 Problem description 1 2 Results 3 3 Subgraphs of positive graphs 6 4 Homomorphic images of positive graphs 10 5 Computational results 14 1 Problem description Let G and H be two simple graphs. A homomorphism G → H is a map V (G) → V (H) that preserves adjacency. We den...

• Type: Journal
• Year: 2016
• Where: EJC
• Authors: Omar Antolín Camarena, Endre Csóka, Tamás Hubai, Gábor Lippner, László Lovász
• Keywords: EJC 2016, Information Technology