Note on 3-paths in plane graphs of girth 4

Abstract

a b s t r a c t An (i, j, k)-path is a path on three vertices u, v and w in this order with deg(u) ≤ i, deg(v) ≤ j, and deg(w) ≤ k. In this paper, we prove that every connected plane graph of girth 4 and minimum degree at least 2 has at least one of the following: a (2, ∞, 2)-path, a (2, 7, 3)-path, a (3, 5, 3)-path, a (4, 2, 5)-path, or a (4, 3, 4)-path. Moreover, no parameter of this description can be improved. Our result supplements recent results concerning the existence of specific 3-paths in plane graphs.

Tight Descriptions of 3‐Paths in Normal Plane Maps

“…Also, Jendrol' and Maceková conjectured that every planar graph with and girth has either a (2, ∞, 2)‐path or . This conjecture was disproved by Aksenov et al.…”

“…The behavior of 3‐paths with low degree sum in sparse planar graphs with was recently studied by Jendrol' and Maceková . Theorem Every planar graph with and girth has a 3‐path of one of the following types: ( i )(2, ∞, 2), (2, 2, 6), (2, 3, 5), (2, 4, 4), or (3, 3, 3) if , ( ii )(2, 2, ∞), (2, 3, 5), (2, 4, 3), or (2, 5, 2) if , ( iii )(2, 2, 6), (2, 3, 3), or (2, 4, 2) if , ( iv )(2, 2, 5) or (2, 3, 3) if , ( v )(2, 2, 3) or (2, 3, 2) if , and ( vi )(2, 2, 2) if . …”

All tight descriptions of 3-paths centered at 2-vertices in plane graphs with girth at least 6

“…Recently, eleven tight descriptions of 3-paths were obtained for δ = 2 and g ≥ 4 by Jendrol', Maceková, Montassier, and Soták [16][17][18][19], four of which descriptions are for g ≥ 9 (for details, see Theorem 2 below).…”