Split Euler Tours In 4-Regular Planar Graphs

• Autorzy: PJ Couch , B.D. Daniel , R. Guidry , W. Paul Wright
• Języki publikacji: EN

Abstrakty

EN

The construction of a homing tour is known to be NP-complete. On the other hand, the Euler formula puts su cient restrictions on plane graphs that one should be able to assert the existence of such tours in some cases; in particular we focus on split Euler tours (SETs) in 3-connected, 4-regular, planar graphs (tfps). An Euler tour S in a graph G is a SET if there is a vertex v (called a half vertex of S) such that the longest portion of the tour between successive visits to v is exactly half the number of edges of G. Among other results, we establish that every tfp G having a SET S in which every vertex of G is a half vertex of S can be transformed to another tfp G′ having a SET S′ in which every vertex of G′ is a half vertex of S′ and G′ has at most one point having a face configuration of a particular class. The various results rely heavily on the structure of such graphs as determined by the Euler formula and on the construction of tfps from the octahedron. We also construct a 2-connected 4-regular planar graph that does not have a SET.

Słowa kluczowe

EN

4-regular; 3-connected; planar; split Euler tour; NP-complete

Kategorie tematyczne

• 05C30: Enumeration in graph theory
• 05C10: Planar graphs; geometric and topological aspects of graph theory
• 05C75: Structural characterization of families of graphs

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2016
• Tom: 36
• Numer: 1
• Strony: 23-30

Daty

wydano

2016-02-01

otrzymano

2013-12-19

poprawiono

2015-02-23

zaakceptowano

2015-03-19

online

2016-01-19

Twórcy

autor

PJ Couch

• Lamar University, Beaumont, Texas, USA

autor

B.D. Daniel

• Lamar University, Beaumont, Texas, USA

autor

R. Guidry

• Austin, Texas, USA

autor

W. Paul Wright

• National Security Agency, USA

Bibliografia

• [1] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (American Elsevier, New York, 1976).
• [2] H.J. Broersma, A.J.W. Duijvestijn and F. Gőbel, Generating all 3-connected 4-regular planar graphs from the octahedron graph, J. Graph Theory 17 (1993) 613–620. doi:10.1002/jgt.3190170508[Crossref]
• [3] J. Czap and S. Jendrol’, Colouring vertices of plane graphs under restrictions given by faces, Discuss. Math. Graph Theory 29 (2009) 521–543. doi:10.7151/dmgt.1462[Crossref]
• [4] N. Dean, Which graphs are pancyclic modulo k?, in: Sixth Intl. Conference on the Theory of Applications of Graphs, (Kalamazoo, Michigan, 1988) 315–326.
• [5] N. Dean, L. Lesniak and A. Saito, Cycles of length 0 modulo 4 in graphs, Discrete Math. 121 (1993) 37–49. doi:10.1016/0012-365X(93)90535-2
• [6] J. Froemke, J.W. Grossman and D.M. Kulkarni, Direct message delivery among processors with limited capacity, in: Graph Theory, Combinatorics, and Algorithms, Vol. 1,2, Kalamazoo, MI, (Wiley-Intersci. Publ., Wiley, New York, 1992) 467–476.
• [7] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979).
• [8] M.R. Garey, D.S. Johnson and R.E. Tarjan, The planar Hamiltonian circuit problem is NP-complete, SIAM J. Comput. 5 (1976) 704–714. doi:10.1137/0205049[Crossref]
• [9] S. Jendrol’ and H.-J. Voss, Light subgraphs of graphs embedded in the plane-A survey, Discrete Math. 313 (2013) 406–421. doi:10.1016/j.disc.2012.11.007[Crossref][WoS]
• [10] H. Lebesgue, Quelques consequences simple de la formula d’Euler, J. Math. Pures Appl. 9 (1940) 27–43.
• [11] J. Lehel, Generating all 4-regular planar graphs from the graph of the octahedron, J. Graph Theory 5 (1981) 423–426. doi:10.1002/jgt.3190050412[Crossref]
• [12] P. Manca, Generating all planar graphs regular of degree four, J. Graph Theory 3 (1979) 357–364. doi:10.1002/jgt.3190030406[Crossref]

Identyfikatory

DOI

10.7151/dmgt.1836