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| Paper IPM / M / 495 |
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Let s′(G) denote the Hall-condition index of a graph G. Hilton and Johnson G is s′-Class 1 if s′(G)=∆(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 ∆(G)=∆(H).
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