Completely Independent Spanning Trees on 4-Regular Chordal Rings

Given a graph G, a set of spanning trees of G are completely independent spanning trees (CISTs for short) if for any vertices x and y, the paths connecting them on these trees have neither vertex nor edge in common, except x and y. Hasunuma (2001, 2002) first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Later on, this conjecture was unfortunately disproved by Péterfalvi (2012). In this note, we show that Hasunuma's conjecture holds for graphs restricted in the class of 4-regular chordal rings CR(n,d), where both n and d are even integers.