Journal of Combinatorial Theory Series B最新文献

For a given graph H, its subdivisions carry the same topological structure. The existence of H-subdivisions within a graph G has deep connections with topological, structural and extremal properties of G. One prominent example of such a connection, due to Bollobás and Thomason and independently Komlós and Szemerédi, asserts that the average degree of G being d ensures a $K_{\Omega \left(\sqrt{d}\right)}$-subdivision in G. Although this square-root bound is best possible, various results showed that much larger clique subdivisions can be found in a graph for many natural classes. We investigate the connection between crux, a notion capturing the essential order of a graph, and the existence of large clique subdivisions. This reveals the unifying cause underpinning all those improvements for various classes of graphs studied. Roughly speaking, when embedding subdivisions, natural space constraints arise; and such space constraints can be measured via crux.

Our main result gives an asymptotically optimal bound on the size of a largest clique subdivision in a generic graph G, which is determined by both its average degree and its crux size. As corollaries, we obtain

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  a characterization of extremal graphs for which the square-root bound above is tight: they are essentially disjoint unions of graphs having crux size linear in d;

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  a unifying approach to find a clique subdivision of almost optimal size in graphs which do not contain a fixed bipartite graph as a subgraph;

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  and that the clique subdivision size in random graphs $G(n,p)$ witnesses a dichotomy: when $p=\omega \left(n^{-1/2}\right)$, the barrier is the space, while when $p=o\left(n^{-1/2}\right)$, the bottleneck is the density.

A graph U is universal for a graph class $C\ni U$, if every $G\in C$ is a minor of U. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or $K_{\infty }$ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that do not have a universal element.

Some of our results and questions may be of interest from the finite graph perspective. In particular, one of our side-results is that every $K_5$-minor-free graph is a minor of a $K_5$-minor-free graph of maximum degree 22.

Pastures are a class of field-like algebraic objects which include both partial fields and hyperfields and have nice categorical properties. We prove several lift theorems for representations of matroids over pastures, including a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields. By embedding the earlier theory into a more general framework, we are able to establish new results even in the case of lifts of partial fields, for example the conjecture of Pendavingh–van Zwam that their lift construction is idempotent. We give numerous applications to matroid representations, e.g. we show that, up to projective equivalence, every pair consisting of a hexagonal representation and an orientation lifts uniquely to a near-regular representation. The proofs are different from the arguments used by Pendavingh and van Zwam, relying instead on a result of Gelfand–Rybnikov–Stone inspired by Tutte's homotopy theorem.

An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group. In this paper we obtain a structure theorem for locally finite quasi-transitive graphs avoiding a minor, which is reminiscent of the Robertson-Seymour Graph Minor Structure Theorem. We prove that every locally finite quasi-transitive graph G avoiding a minor has a tree-decomposition whose torsos are finite or planar; moreover the tree-decomposition is canonical, i.e. invariant under the action of the automorphism group of G. As applications of this result, we prove the following.

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  Every locally finite quasi-transitive graph attains its Hadwiger number, that is, if such a graph contains arbitrarily large clique minors, then it contains an infinite clique minor. This extends a result of Thomassen (1992) [38] who proved it in the (quasi-)4-connected case and suggested that this assumption could be omitted. In particular, this shows that a Cayley graph excludes a finite minor if and only if it avoids the countable clique as a minor.

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  Locally finite quasi-transitive graphs avoiding a minor are accessible (in the sense of Thomassen and Woess), which extends known results on planar graphs to any proper minor-closed family.

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  Minor-excluded finitely generated groups are accessible (in the group-theoretic sense) and finitely presented, which extends classical results on planar groups.

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  The domino problem is decidable in a minor-excluded finitely generated group if and only if the group is virtually free, which proves the minor-excluded case of a conjecture of Ballier and Stein (2018) [7].