Abstract
Recent work of Beraha, Hall, and Tutte has made it clear that it is interesting to express the constrained chromials of planar graphs containing an open n-circuit in terms of free chromials. This problem was solved for n = 4 and n = 5, and a partial solution given for n = 6 by Birkhoff and Lewis. The solution for the case n = 6 was completed by Hall and Lewis. There are 162 constrained chromials for the seven circuit. One of these has all 7 colors different, the remaining 161 divide themselves into 23 groups of seven each. The sum of the constrained chromials in each of these 23 groups is called a constrained rochromial. The sum of the corresponding 7 free chromials in each group is called a free rochromial. Equations are obtained expressing each of the constrained rochromials in terms of free rochromials. A new Beraha number appears exactly where it should.
This paper was partially supported by grants from the SUNY Research Foundation
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References
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© 1974 Springer-Verlag Berlin
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Hall, D.W. (1974). Coloring seven-circuits. In: Bari, R.A., Harary, F. (eds) Graphs and Combinatorics. Lecture Notes in Mathematics, vol 406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066449
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DOI: https://doi.org/10.1007/BFb0066449
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