The Hall-Condition Index of a Graph and the Overfull Conjecture

Let \(s'(G)\) denote the Hall-condition index of a graph \(G\). Hilton and Johnson recently introduced this parameter and proved that \(\Delta(G) \leq s'(G) \leq \Delta(G) + 1\). A graph \(G\) is \(s’\)-Class 1 if \(s'(G) = \Delta(G)\) and is \(s’\)-Class 2 otherwise. A graph \(G\) is \(s’\)-critical if \(G\) is connected, \(s’\)-Class 2, and, for every edge \(e\), \(s'(G – e) < s'(G)\). We use the concept of the fractional chromatic index of a graph to classify \(s’\)-Class 2 in terms of overfull subgraphs, and similarly to classify \(s’\)-critical graphs. We apply these results to show that the following variation of the Overfull Conjecture is true;

A graph \(G\) is \(s’\)-Class 2 if and only if \(G\) contains an overfull subgraph \(H\) with \(\Delta(G) = \Delta(H)\).