Abstract
Grünbaum and Malkevitch proved that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most 76/77. Recently, this was improved to 359/366 (< 52/53) and the question was raised whether this can be strengthened to 41/42, a natural bound inferred from one of the Faulkner-Younger graphs. We prove that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most 37/38 and that we also get the same value for cyclically 4-edge-connected cubic graphs of genus g for any prescribed genus g ≥ 0. We also show that 45/46 is an upper bound for the shortness coefficient of cyclically 4-edge-connected cubic graphs of genus g with face lengths bounded above by some constant larger than 22 for any prescribed g ≥ 0.
Highlights
We prove that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most
In 1973, Grunbaum and Walther [12] introduced two limits called shortness coefficient and shortness exponent that measure how far a given infinite family G of graphs is from being Hamiltonian
Tutte’s celebrated result that 4-connected planar graphs are Hamiltonian [26] implies that the shortness coefficient of the 4-connected planar graphs is 1, and the same conclusion holds if we relax the prerequisite of 4-connectedness to ‘containing at most three 3-vertex-cuts’ [4]—for a more detailed overview of hamiltonicity in planar graphs with few 3-vertex-cuts we refer the reader to [22]
Summary
In 1973, Grunbaum and Walther [12] introduced two limits called shortness coefficient and shortness exponent that measure how far a given infinite family G of graphs is from being Hamiltonian. Tutte’s celebrated result that 4-connected planar graphs are Hamiltonian [26] implies that the shortness coefficient of the 4-connected planar graphs is 1, and the same conclusion holds if we relax the prerequisite of 4-connectedness to ‘containing at most three 3-vertex-cuts’ [4]—for a more detailed overview of hamiltonicity in planar graphs with few 3-vertex-cuts we refer the reader to [22] It is well-known that infinitely many non-Hamiltonian graphs appear when sufficiently many 3-vertex-cuts are present: Moon and Moser [20] showed that the shortness exponent of the 3-connected planar (and even maximal planar) graphs is at most log , while Chen and Yu [5] showed that this upper bound is tight, i.e. the shortness exponent of these graphs is log . Despite the lack of non-trivial lower bounds for ρ(C4), an upper bound for ρ(C4) is known: Maˇcajova and Mazak [19] showed recently that C4 contains an infinite graph family in which the circumference of every n-vertex graph. We will make tacit use of the Jordan Curve Theorem
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