a b s t r a c tUsing topological circles in the Freudenthal compactification of a graph as infinite cycles, we extend to locally finite graphs a result of Oberly and Sumner on the Hamiltonicity of finite graphs. This answers a question of Stein, and gives a sufficient condition for Hamiltonicity in locally finite graphs.

Among the well-known sufficient degree conditions for the Hamiltonicity of a finite graph, the condition of Asratian and Khachatrian is the weakest and thus gives the strongest result. Diestel conjectured that it should extend to locally finite infinite graphs G, in that the same condition implies that the Freudenthal compactification of G contains a circle through all its vertices and ends. We prove Diestel's conjecture for claw-free graphs.

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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.

A dicut in a directed graph is a cut for which all of its edges are directed to a common side of the cut. A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of an edge set meeting every dicut equals the maximum number of disjoint dicuts in that digraph. In this first paper out of a series of two papers, we conjecture a version of this theorem using a more structural description of this min-max property for finite dicuts in infinite digraphs.We show that this conjecture can be reduced to countable digraphs where the underlying undirected graph is 2-connected, and we prove several special cases of the conjecture.

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In this paper we characterise the graphs containing a Z × Z grid minor in a similar way as it has been done by Halin for graphs with an N × Z grid minor. Using our characterisation, we describe the structure of graphs without Z × Z grid minors in terms of tree-decompositions.