• Corpus ID: 59405451

Spanning closed walks with bounded maximum degrees of graphs on surfaces

@article{Hasanvand2017SpanningCW,
  title={Spanning closed walks with bounded maximum degrees of graphs on surfaces},
  author={Morteza Hasanvand},
  journal={arXiv: Combinatorics},
  year={2017}
}
Gao and Richter (1994) showed that every $3$-connected graph which embeds on the plane or the projective plane has a spanning closed walk meeting each vertex at most $2$ times. Brunet, Ellingham, Gao, Metzlar, and Richter (1995) extended this result to the torus and Klein bottle. Sanders and Zhao (2001) obtained a sharp result for higher surfaces by proving that every $3$-connected graph embeddable on a surface with Euler characteristic $\chi \le -46$ admits a spanning closed walk meeting each… 
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References

SHOWING 1-10 OF 19 REFERENCES

On spanning trees and walks of low maximum degree

This article uses the discharging method to obtain the best possible results that a 3-connected graph embeddable on a surface of Euler characteristic χ ≤ -46 has a spanning tree of maximum degree at…

Spanning Planar Subgraphs of Graphs in the Torus and Klein Bottle

It is shown that every 3-connected graph embedded in the torus or the Klein bottle has a spanning planar subgraph which is 2-connected, and in fact has a slightly stronger connectivity property.

Spanning Trees with Bounded Maximum Degrees of Graphs on Surfaces

  • K. Ozeki
  • Mathematics
    SIAM J. Discret. Math.
  • 2013
If G is a $3-connected graph on a surface with Euler characteristic $\chi < 0$, then G has a spanning $\lceil\frac{8-2\chi}{3}\rceil$-tree $T$ with $te(T, 3) \leq - 2\chi-1$.

2-Walks in Circuit Graphs

Abstract We prove the conjecture of Jackson and Wormald that every 3-connected planar graph has a closed walk visiting every vertex once or twice. This strengthens Barnette′s Theorem that every…

Spanning trees and spanning Eulerian subgraphs with small degrees

Spanning eulerian subgraphs of bounded degree in triangulations

It is shown that all triangulations in the projective plane, the torus and the Klein bottle have spanning Eulerian subgraphs with maximum degree at most eight.

On 2-Connected Spanning Subgraphs with Low Maximum Degree

It is shown that a 4-connected graph embeddable on a surface with non-negative Euler characteristic has a 3-trestle, approaching a conjecture of Nash-Williams.

Double cycle covers and the petersen graph

It is shown that if a graph G is at most 5 edges short of being 4-edge-connected, then exactly one of these holds: G ϵ 3, G has at least one cut-edge, or G is contractible to the Petersen graph.

Spanning trees in 3-connected K3, t-minor-free graphs

K-walks of Graphs

This work generalises the result of Oberly and Sumner that every connected, locally connected K l,3-free graph with at least three vertices is hamil tonian.