We present an algorithm to 3-colour a graph G without triangles or induced paths on seven vertices in O(|V (G)| 7 ) time. In fact, our algorithm solves the list 3-colouring problem, where each vertex is assigned a subset of {1, 2, 3} as its admissible colours.
We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel and Kühn [4,5], which allows for infinite cycles, we prove that the edge set of a locally finite graph G lies in C(G) if and only if every vertex and every end has even degree. In the same way we generalise to locally finite graphs the characterisation of the cycles in a finite graph as its 2-regular connected subgraphs.
Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph.As an application, we show that the k-blocks -the maximal vertex sets that cannot be separated by at most k vertices -of a graph G live in distinct parts of a suitable tree-decomposition of G of adhesion at most k, whose decomposition tree is invariant under the automorphisms of G. This extends recent work of Dunwoody and Krön and, like theirs, generalizes a similar theorem of Tutte for k = 2.Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all k simultaneously, all the k-blocks of a finite graph.
Loebl, Komlós, and Sós conjectured that if at least half of the vertices of a graph G have degree at least some k ∈ N, then every tree with at most k edges is a subgraph of G. Our main result is an approximate version of this conjecture for large enough n = |V (G)|, assumed that n = O(k).Our result implies an asymptotic bound for the Ramsey number of trees. We prove that r(T k , Tm) ≤ k + m + o(k + m), as k + m → ∞.
MacLane's planarity criterion states that a finite graph is planar if and only if its cycle space has a basis B such that every edge is contained in at most two members of B. Solving a problem of Wagner [Graphentheorie, Bibliographisches Institut, Mannheim, 1970], we show that the topological cycle space introduced recently by Diestel and Kühn allows a verbatim generalisation of MacLane's criterion to locally finite graphs. This then enables us to extend Kelmans' planarity criterion as well.
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