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].

Graphings serve as limit objects for bounded-degree graphs. We define the “cycle matroid” of a graphing as a submodular setfunction, with values in $[0,1]$, which generalizes (up to normalization) the cycle matroid of finite graphs. We prove that for a Benjamini–Schramm convergent sequence of graphs, the total rank, normalized by the number of nodes, converges to the total rank of the limit graphing.

Let G and H be hypergraphs on n vertices, and suppose H has large enough minimum degree to necessarily contain a copy of G as a subgraph. We give a general method to randomly embed G into H with good “spread”. More precisely, for a wide class of G, we find a randomised embedding $f:G\hookrightarrow H$ with the following property: for every s, for any partial embedding $f^{\prime }$ of s vertices of G into H, the probability that f extends $f^{\prime }$ is at most $O{(1/n)}^s$. This is a common generalisation of several streams of research surrounding the classical Dirac-type problem.

For example, setting $s=n$, we obtain an asymptotically tight lower bound on the number of embeddings of G into H. This recovers and extends recent results of Glock, Gould, Joos, Kühn, and Osthus and of Montgomery and Pavez-Signé regarding enumerating Hamilton cycles in Dirac hypergraphs. Moreover, using the recent developments surrounding the Kahn–Kalai conjecture, this result implies that many Dirac-type results hold robustly, meaning G still embeds into H after a random sparsification of its edge set. This allows us to recover a recent result of Kang, Kelly, Kühn, Osthus, and Pfenninger and of Pham, Sah, Sawhney, and Simkin for perfect matchings, and obtain novel results for Hamilton cycles and factors in Dirac hypergraphs.

Notably, our randomised embedding algorithm is self-contained and does not require Szemerédi's regularity lemma or iterative absorption.

In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal–Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a 3-edge-colored graph with R red, G green, B blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$, which is sharp. Second, we give a generalization of the Kruskal–Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.

Given a finite simple graph Γ on n vertices its complementary prism is the graph $\Gamma \overline{\Gamma }$ that is obtained from Γ and its complement $\overline{\Gamma }$ by adding a perfect matching where each its edge connects two copies of the same vertex in Γ and $\overline{\Gamma }$. It generalizes the Petersen graph, which is obtained if Γ is the pentagon. The automorphism group of $\Gamma \overline{\Gamma }$ is described for an arbitrary graph Γ. In particular, it is shown that the ratio between the cardinalities of the automorphism groups of $\Gamma \overline{\Gamma }$ and Γ can attain only the values 1, 2, 4, and 12. It is shown that $\Gamma \overline{\Gamma }$ is vertex-transitive if and only if Γ is vertex-transitive and self-complementary. Moreover, the complementary prism is not a Cayley graph whenever $n>1$.

Let H be an h-vertex graph. The vertex arboricity $ar\left(H\right)$ of H is the least integer r such that $V\left(H\right)$ can be partitioned into r parts and each part induces a forest in H. We show that for sufficiently large $n\in hN$, every n-vertex graph G with $\delta \left(G\right)\ge \max \{(1-\frac{2}{f\left(H\right)}+o(1\left)\right)n,(\frac{1}{2}+o(1\left)\right)n\}$ and $\alpha \left(G\right)=o\left(n\right)$ contains an H-factor, where $f\left(H\right)=2ar\left(H\right)$ or $2ar\left(H\right)-1$. The result can be viewed an analogue of the Alon–Yuster theorem [1] in Ramsey–Turán theory, which generalizes the results of Balogh–Molla–Sharifzadeh [2] and Knierim–Su [21] on clique factors. In particular the degree conditions are asymptotically sharp for infinitely many graphs H which are not cliques.