×

5-regular 3-polytopal graphs with edges of only two types and shortness exponents less than one. (English) Zbl 0852.05056

This paper shows that if \(q\geq 29\) and \(q\not\equiv 0\pmod 3\), then the infinite class of 5-regular 3-polytopal graphs whose edges are incident with either two triangles or a triangle and a \(q\)-gon contains non-Hamiltonian members and even has shortness exponent less than one.

MSC:

05C35 Extremal problems in graph theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Grünbaum, B.; Walther, H., Shortness exponents of families of graphs, J. Combin. Theory Ser. A, 14, 364-385 (1973) · Zbl 0263.05103
[2] J. Harant, Über den shortness exponent regulärer Polyedergraphen mit genau zwei Typen von Elementarflächen, Thesis A, Ilmenau Institute of Technology.; J. Harant, Über den shortness exponent regulärer Polyedergraphen mit genau zwei Typen von Elementarflächen, Thesis A, Ilmenau Institute of Technology.
[3] Jendroľ, S.; Kekeňák, R., Longest circuits in triangular and quadrangular 3-polytopes with two types of edges, Math. Slovaca, 40 (1990) · Zbl 0757.05073
[4] S. Jendroľ and P.J. Owens, Pentagonal 3-polytopal graphs with edges of only two types and shortness parameters, Discrete Math., to appear.; S. Jendroľ and P.J. Owens, Pentagonal 3-polytopal graphs with edges of only two types and shortness parameters, Discrete Math., to appear.
[5] Jendroľ, S.; Tkáľ, M., Convex polytopes with exactly two types of edges, Discrete Math., 84, 143-160 (1990) · Zbl 0705.52014
[6] Owens, P. J., Shortness parameters of families of regular planar graphs with two or three types of faces, Discrete Math., 39, 199-209 (1982) · Zbl 0492.05051
[7] Owens, P. J., Regular planar graphs with faces of only two types and shortness parameters, J. Graph Theory, 8, 253-275 (1984) · Zbl 0541.05037
[8] Tutte, W. T., On Hamiltonian circuits, J. London Math. Soc., 21, 98-101 (1946) · Zbl 0061.41306
[9] Walther, H., Über das Problem der Existenz von Hamiltonkreisen in planaren, regulären Graphen, Math. Nachr., 39, 277-296 (1969) · Zbl 0169.26401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.