Journal of Combinatorial Theory Series B最新文献

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained. This is the first complete result about obstructions for embeddability of graphs in the Klein bottle, and the outcome is somewhat surprising in the sense that there are considerably fewer excluded minors than expected.

We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result [20] of Mančinska and Roberson shows that graphs G and H are quantum isomorphic if and only if, for any planar graph F, the number of graph homomorphisms from F to G is equal to the number of graph homomorphisms from F to H. A generalization of partition functions called “scaffolds” [23] affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes. The paper includes open problems and directions for future research.

Let us say a graph is $O_s$-free, where $s\ge 1$ is an integer, if there do not exist s cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when $s=2$, is not well understood. For instance, until now we did not know how to test whether a graph is $O_2$-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that $O_2$-free graphs have only a polynomial number of induced paths.

In this paper we prove Le's conjecture; indeed, we will show that for all $s\ge 1$, there exists $c>0$ such that every $O_s$-free graph G has at most $|G{|}^c$ induced paths, where $\left|G\right|$ is the number of vertices. This provides a poly-time algorithm to test if a graph is $O_s$-free, for all fixed s.

The proof has three parts. First, there is a short and beautiful proof, due to Le, that reduces the question to proving the same thing for graphs with no cycles of length four. Second, there is a recent result of Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek, that in every $O_s$-free graph G with no cycle of length four, there is a set of vertices that intersects every cycle, with size logarithmic in $\left|G\right|$. And third, there is an argument that uses the result of Bonamy et al. to deduce the theorem. The last is the main content of this paper.

Applying a recent extension (2015) of a structure theorem of Robertson, Seymour and Thomas from 1993, in this paper we establish Robertson's magic-tree conjecture from 1997.

The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. Babai conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree $\ge cn$ for some constant $c>0$. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3.

In this paper, we extend Babai's result to primitive coherent configurations of rank 4, confirming the conjecture in this special case. The proofs combine structural and spectral methods.

Recently (March 2022) Sean Eberhard published a class of counterexamples of rank 28 to Babai's conjecture and suggested to replace “Cameron schemes” in the conjecture with a more general class he calls “Cameron sandwiches”. Naturally, our result also confirms the rank 4 case of Eberhard's version of the conjecture.

We prove that for every $\epsilon >0$ there exists $n_0=n_0\left(\epsilon \right)$ such that every regular oriented graph on $n>n_0$ vertices and degree at least $(1/4+\epsilon )n$ has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.

The notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999.

In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introduce a version of Erdős-Pósa property for matching minors and find a direct link between this property and planarity. From this, it follows that a class of bipartite graphs with perfect matchings has bounded perfect matching width if and only if it excludes a planar matching minor. We also present algorithms for bipartite graphs of bounded perfect matching width for a matching version of the disjoint paths problem, matching minor containment, and for counting the number of perfect matchings. From our structural results, we obtain that recognising whether a bipartite graph G contains a fixed planar graph H as a matching minor, and that counting the number of perfect matchings of a bipartite graph that excludes a fixed planar graph as a matching minor are both polynomial time solvable.

In this paper, we prove that for any $k\ge 3$, there exist infinitely many minimal asymmetric k-uniform hypergraphs. This is in a striking contrast to $k=2$, where it has been proved recently that there are exactly 18 minimal asymmetric graphs.

We also determine, for every $k\ge 1$, the minimum size of an asymmetric k-uniform hypergraph.

A theorem of Mader shows that every graph with average degree at least eight has a $K_6$ minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have $K_6$ minors, but minimum degree six is certainly not enough. For every $\epsilon >0$ there are arbitrarily large graphs with average degree at least $8-\epsilon$ and minimum degree at least six, with no $K_6$ minor.

But what if we restrict ourselves to bipartite graphs? The first statement remains true: for every $\epsilon >0$ there are arbitrarily large bipartite graphs with average degree at least $8-\epsilon$ and no $K_6$ minor. But surprisingly, going to minimum degree now makes a significant difference. We will show that every bipartite graph with minimum degree at least six has a $K_6$ minor. Indeed, it is enough that every vertex in the larger part of the bipartition has degree at least six.

A family of subsets of $\left[n\right]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl–Kupavskii and Balogh–Das–Liu–Sharifzadeh–Tran showed that for $n\ge 2k+c\sqrt{k\ln k}$, almost all k-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n\ge 2k+100\ln k$. Our proof uses, among others, the graph container method and the Das–Tran removal lemma.