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Light paths with an odd number of vertices in polyhedral maps

Stanislav Jendroľ, Heinz-Jürgen Voss (2000)

Czechoslovak Mathematical Journal

Let P k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2 -manifold 𝕄 with Euler characteristic χ ( 𝕄 ) ≤ 0 contains a path P k such that each vertex of this path has, in G , degree ≤ k 5 + 49 - 24 χ ( 𝕄 ) 2 . Moreover, this bound is attained for k = 1 or k ≥ 2 , k even. In this paper we prove that for each odd k ≥ 4 3 5 + 49 - 24 χ ( 𝕄 ) 2 + 1 , this bound is the best possible on infinitely many compact 2 -manifolds, but on infinitely many other compact 2 -manifolds the upper bound can be lowered to ( k - 1 3 ) 5 + 49 - 24 χ ( 𝕄 ) 2 .

Link homotopy invariants of graphs in R3.

Kouki Taniyama (1994)

Revista Matemática de la Universidad Complutense de Madrid

In this paper we define a link homotopy invariant of spatial graphs based on the second degree coefficient of the Conway polynomial of a knot.

Looseness and Independence Number of Triangulations on Closed Surfaces

Atsuhiro Nakamoto, Seiya Negami, Kyoji Ohba, Yusuke Suzuki (2016)

Discussiones Mathematicae Graph Theory

The looseness of a triangulation G on a closed surface F2, denoted by ξ (G), is defined as the minimum number k such that for any surjection c : V (G) → {1, 2, . . . , k + 3}, there is a face uvw of G with c(u), c(v) and c(w) all distinct. We shall bound ξ (G) for triangulations G on closed surfaces by the independence number of G denoted by α(G). In particular, for a triangulation G on the sphere, we have [...] and this bound is sharp. For a triangulation G on a non-spherical surface F2, we have...

Minimal vertex degree sum of a 3-path in plane maps

O.V. Borodin (1997)

Discussiones Mathematicae Graph Theory

Let wₖ be the minimum degree sum of a path on k vertices in a graph. We prove for normal plane maps that: (1) if w₂ = 6, then w₃ may be arbitrarily big, (2) if w₂ < 6, then either w₃ ≤ 18 or there is a ≤ 15-vertex adjacent to two 3-vertices, and (3) if w₂ < 7, then w₃ ≤ 17.

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