29 research outputs found
Pairwise disjoint perfect matchings in -edge-connected -regular graphs
Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B,
141 (2020), 343-351] asked whether every -edge-connected -regular graph
of even order has pairwise disjoint perfect matchings. We show that this
is not the case if . Together with a recent result
of Mattiolo and Steffen [Highly edge-connected regular graphs without large
factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves
Thomassen's problem for all even . It turns out that our methods are limited
to the even case of Thomassen's problem. We then prove some equivalences of
statements on pairwise disjoint perfect matchings in highly edge-connected
regular graphs, where the perfect matchings contain or avoid fixed sets of
edges. Based on these results we relate statements on pairwise disjoint perfect
matchings of 5-edge-connected 5-regular graphs to well-known conjectures for
cubic graphs, such as the Fan-Raspaud Conjecture, the Berge-Fulkerson
Conjecture and the -Cycle Double Cover Conjecture.Comment: 24 page
Every graph occurs as an induced subgraph of some hypohamiltonian graph
We prove the titular statement. This settles a problem of Chvátal from 1973 and encompasses earlier results of Thomassen, who showed it for K_3, and Collier and Schmeichel, who proved it for bipartite graphs. We also show that for every outerplanar graph there exists a planar hypohamiltonian graph containing it as an induced subgraph
CONJECTURE JACKSON PADA SUBGRAPH EULERIAN
Permasalahan dalam skripsi ini adalah bagaimana hubungan antara conjecture Jackson dan conjecture Thomassen. Untuk membahas hubungan tersebut digunakan metode analisis dari teorema-teorema dalam graph hamilton, graph garis, dan graph-graph yang memenuhi conjecture Jackson dan conjecture Thomassen
Disjoint isomorphic balanced clique subdivisions
A classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least has a subdivision of , the complete graph on k vertices. We study two directions extending this result. • Verstraëte conjectured that a quadratic bound guarantees in fact two vertex-disjoint isomorphic copies of a -subdivision.
• Thomassen conjectured that for each there is some such that every graph with average degree at least d contains a balanced subdivision of . Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on remains open. In this paper, we show that a quadratic lower bound on average degree suffices to force a balanced -subdivision. This gives the right order of magnitude of the optimal needed in Thomassen's conjecture. Since a balanced -subdivision trivially contains m vertex-disjoint isomorphic -subdivisions, this also confirms Verstraëte's conjecture in a strong sense
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Polyhedra of small order and their Hamiltonian properties
We describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up to 12 vertices and up to 26 edges, simplical polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, and bipartite polyhedra with up to 22 vertices.The results of the enumeration were used to systematically search for certain minimal non-Hamiltonian polyhedra. In particular, the smallest polyhedra satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given
Chords of longest circuits of graphs
This thesis is on a long standing open conjecture proposed by one of the most prominent mathematicians, Dr. C. Thomassen: Every longest circuit of 3-connected graph has a chord. In 1987, C. Q. Zhang proved that every longest circuit of a 3-connected planar graph G has a chord if G is cubic or if the minimum degree is at least 4. In 1997, Carsten Thomassen proved that every longest circuit of 3-connected cubic graph has a chord.;In this dissertation, we prove the following three independent partial results: (1) Every longest circuit of a 3-connected graph embedded in a projective plane with minimum degree at least has a chord (Theorem 2.3.1). (2) Every longest circuit of a 3-connected cubic graph has at least two chords. Furthermore if the graph is also a planar, then every longest circuit has at least three chords (Theorem 3.2.6, 3.2.7). (3) Every longest circuit of a 4-connected graph embedded in a torus or Klein bottle has a chord.;We get these three independent results with three totally different approaches: Connectivity (Tutte circuit), second Hamilton circuit, and charge and discharge methods