Local Ore Conditions for Rainbow Pancyclicity and Vertex-Pancyclicity in Graph Systems

Abstract

We investigate rainbow pancyclicity and vertex-pancyclicity of graph systems satisfying a local Ore-type degree condition. All graphs in the system share the same vertex set. We prove that if each graph is 2-connected and satisfies our local Ore condition, then the graph system is either rainbow pancyclic and vertex-pancyclic for cycles of length from 4 to n, or all graphs are identical balanced complete bipartite graphs on an even vertex set. A stronger degree threshold ensures full rainbow vertex-pancyclicity. All thresholds we obtain are tight. Our proofs rely on cycle extension techniques and neighborhood separation arguments.