Subcubic Edge‐Chromatic Critical Graphs Have Many Edges

Abstract

We consider graphs G with ∆ = 3 such that χ ′ (G) = 4 and χ ′ (G − e) = 3 for every edge e, so-called critical graphs. Jakobsen noted that the Petersen graph with a vertex deleted, P * , is such a graph and has average degree only 2 + 2 3 . He showed that every critical graph has average degree at least 2 + 2 3 , and asked if P * is the only graph where equality holds. We answer his question affirmatively. Our main result is that every subcubic critical graph, other than P * , has average degree at least 2 + 26 37 = 2.702.

The average degree of a subcubic edge‐chromatic critical graph

“…Conjecture (1) states that every 3‐critical graph has average degree at least ; as mentioned earlier, this was proved by Jakobsen . Cranston and Rabern recently proved Conjecture (2), which states that if is a 3‐critical graph different from , then . Using a computer, they also made an attempt on Conjecture (3), proving that if is 3‐critical and not nor of the form , then .…”

“…Let . The following result was proved in Cranston and Rabern , Lemma 5.…”

“…We recall the following simple result, which is well known and easy to prove (see Cranston and Rabern , Lemma 4).…”

“…Lemma (a) implies that every vertex of has degree 2 or 3. Following Cranston and Rabern , we call a 3‐vertex of poor if it has a 2‐neighbor, and rich otherwise. The poor subgraph of is the subgraph induced by the poor vertices; its components are the poor fragments of .…”

“…Clearly Conjecture implies Conjecture , for each . But the argument of (Cranston and Rabern , Claims 1 and 3 of Lemma 6) can be modified to show that a minimal counterexample to Conjecture , for any fixed , is triangle‐free; thus Conjecture is equivalent to Conjecture , and Conjecture is equivalent to Conjecture .…”