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Hypohamiltonian snarks. (English) Zbl 0535.05045
Graphs and other combinatorial topics, Proc. 3rd Czech. Symp., Prague 1982, Teubner-Texte Math. 59, 70-75 (1983).
[For the entire collection see Zbl 0517.00002.]
A graph G is hypohamiltonian if G is not Hamiltonian but G-v is Hamiltonian for all $$v\in V(G)$$. A snark is a cyclically 4-edge connected cubic graph with girth at least 5 that is not Tait-colorable (and thus non-Hamiltonian). The author shows that R. Isaacs’ flower snarks [Am. Math. Mon. 82, 221-239 (1975; Zbl 0311.05109)] are hypohamiltonian but that Isaacs’ construction for generating a snark from two smaller snarks produces a hypohamiltonian snark from two smaller hypohamiltonian snarks only under special conditions. He states that the M. K. Gol’dberg [J. Comb. Theory, Ser. B 31, 282-291 (1981; Zbl 0449.05037)] are not hypohamiltonian and relays Chetwynd’s result that the double-star snark of Isaacs on 30 vertices is hypohamiltonian.
Misprints in the paths P sequences on page 72 should be corrected as follows. $$F_{4j+3}-a_ 5:$$ change the first $$a_ 6$$ to $$a_ 7$$, the second $$c_ 7$$ to $$c_ 6$$ and d to $$d_ 7$$. $$F_{4j+3}-c_ 3:$$ transpose $$d_ 2$$ and $$b_ 2$$. $$F_{4j+1}-c_ 3:$$ change the first $$a_ 3$$ to $$d_ 3$$.
Reviewer: R.Entringer

##### MSC:
 05C45 Eulerian and Hamiltonian graphs