Inner Local Exponent of A Two-cycle Non-Hamiltonian Two-coloured Digraph with Cycle Lengths n and 3n + 1

A digraph that has arcs of two colours is called a two-coloured digraph. In this case, the colours used are red and black. Let d and k be non-negative integers, where d represents the number of red arcs and k represents the number of black arcs. A (d, k)-walk on the two-coloured digraph is defined as a walk with d red arcs and k black arcs. The smallest integer sum of d and k such that there is a (d, k)-walk from vertex y to vertex z is called the exponent number of two-coloured digraph, whereas the smallest integer sum of d and k such that there is (d, k)-walk from each vertex to vertex vx is called the inner local exponent of a vertex vx. This article discusses the inner local exponent of a two-cycle non-Hamiltonian two-coloured digraph with cycle lengths n and 3n+1. This digraph has exactly four red arcs. The four red arcs are combined consecutively or alternately when there is one allied vertex. © (2024), (International Association of Engineers). All rights reserved.