Hamiltonicity in Locally Finite Graphs: Two Extensions and a Counterexample

Abstract

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs.We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing K 4 or K 2,3 as a minor is Hamiltonian if and only if it is 2-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and formed by the 2-contractible edges.The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist. arXiv:1701.06029v2 [math.CO] 5 Dec 2018From this we then obtain the following corollary, which extends statement (ii) of Proposition 1.6. Corollary 1.9. Let G be a locally finite 2-connected graph not containing K 4 or K 2,3 as a minor, and not isomorphic to K 3 . Then the edges contained in the Hamilton circle of |G| are precisely the 2-contractible edges of G.We should note here that parts of Theorem 1.8 and Corollary 1.9 are already known. Chan [5, Thm. 20 with Thm. 27] proved that a locally finite 2-connected graph not isomorphic to K 3 and not containing K 4 or K 2,3 as a minor has a Hamilton circle that consists precisely of the 2-contractible edges of the graph. He deduces this from other general results about 2-contractible edges in locally finite 2connected graphs. In our proof, however, we directly construct the Hamilton circle and show its uniqueness without working with 2-contractible edges. Afterwards, we deduce Corollary 1.9.Our third result is related to the following conjecture Sheehan made for finite graphs.Conjecture 1.10.[26] There is no finite r-regular graph with a unique Hamilton cycle for any r > 2.

Hamilton Cycles in Infinite Cubic Graphs

“…Let T ′ = T /{ℓ x = ℓ y = ℓ z } be the graph obtained from T by identifying its three leaves. Then T ′ is a cubic graph with precisely 6 Hamilton cycles (see [2,6,11]), which we may visualise as follows: The first two Hamilton cycles use the edge pair e x = ℓ x x and e z = ℓ z z, and the other four Hamilton cycles use the edge pair e y = ℓ y y and e z . In particular, there are no Hamilton cycles of T ′ using the edge pair {e x , e y }.…”

“…Does there exist a uniquely Hamiltonian, d-regular graph for d ≥ 3 where also all ends have degree d? K. Heuer [6] has recently constructed a uniquely Hamiltonian cubic graph with continuum many ends where all ends have degree 3, thus answering Question 1. 5.…”

Hamiltonicity in infinite tournaments

“…Letting topological arcs and circles in |G| take the role of paths and cycles in G, it often becomes possible to extend theorems about paths and cycles in finite graphs to infinite graphs. Examples include Euler's theorem [11,20], arboricity and tree-packing [7,27], Hamiltonicity [9,12,13,15,16,17,23], and various planarity criteria [1,14,24].…”

Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs I: Nets and bulls

“…This definition now allows a rather big variety of infinite cycles. We call a circle a Hamilton circle of G if it contains all vertices of G. Since Hamilton circles are closed subspaces of   G , they also contain all ends of G. Several Hamiltonicity results have been extended to locally finite infinite graphs so far, although not always completely, but with additional requirements [2,3,[12][13][14][15][16][17][18][20][21][22][23]. For finite graphs many sufficient conditions guaranteeing Hamiltonicity exist which make use of global assumptions for example degree conditions involving the total number of vertices.…”