Journal of Combinatorial Theory Series B最新文献

Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the “middle ranges”, the notion of subgraph densities in these limit objects remains elusive. We define subgraph densities in the orthogonality graphs on the unit spheres in dimension d, under an appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs in the “middle” range, and so we expect their study to serve as a key example to defining subgraph densities in more general Markov spaces. Interest in studying homomorphisms of a finite graph G into orthogonality graphs is supported by the fact that such homomorphisms are just the orthonormal representations of the complementary graph.

We continue the study of $(\mathrm{tw},\omega )$-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this property has useful algorithmic implications for the Maximum Independent Set and related problems.

In the previous paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. II. Tree-independence number, J. Comb. Theory, Ser. B, 164 (2024) 404–442], we introduced the tree-independence number, a min-max graph invariant related to tree decompositions. Bounded tree-independence number implies both $(\mathrm{tw},\omega )$-boundedness and the existence of a polynomial-time algorithm for the Maximum Weight Independent Packing problem, provided that the input graph is given together with a tree decomposition with bounded independence number. In particular, this implies polynomial-time solvability of the Maximum Weight Independent Set problem.

In this paper, we consider six graph containment relations—the subgraph, topological minor, and minor relations, as well as their induced variants—and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. The induced minor relation is of particular interest: we show that excluding either a $K_5$ minus an edge or the 4-wheel implies the existence of a tree decomposition in which every bag is a clique plus at most 3 vertices, while excluding a complete bipartite graph $K_{2,q}$ implies the existence of a tree decomposition with independence number at most $2(q-1)$.

These results are obtained using a variety of tools, including ℓ-refined tree decompositions, SPQR trees, and potential maximal cliques, and actually show that in the bounded cases identified in this work, one can also compute tree decompositions with bounded independence number efficiently. Applying the algorithmic framework provided by the previous paper in the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. In particular, these results apply to the class of 1-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius from 2019.

A triple system is cancellative if it does not contain three distinct sets $A,B,C$ such that the symmetric difference of A and B is contained in C. We show that every cancellative triple system $H$ that satisfies a particular inequality between the sizes of $H$ and its shadow must be structurally close to the balanced blowup of some Steiner triple system. Our result contains a stability theorem for cancellative triple systems due to Keevash and Mubayi as a special case. It also implies that the boundary of the feasible region of cancellative triple systems has infinitely many local maxima, thus giving the first example showing this phenomenon.

We prove an arithmetic analogue of the typical structure theorem for graph hereditary properties due to Alon, Balogh, Bollobás and Morris [2].

A graph is $O_k$-free if it does not contain k pairwise vertex-disjoint and non-adjacent cycles. We prove that “sparse” (here, not containing large complete bipartite graphs as subgraphs) $O_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of $O_2$-free graphs without $K_{2,3}$ as a subgraph and whose treewidth is (at least) logarithmic.

Using our result, we show that Maximum Independent Set and 3-Coloring in $O_k$-free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse $O_k$-free graphs, and that deciding the $O_k$-freeness of sparse graphs is polynomial time solvable.

In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond percolation is applied. Far from being able to classify all such degree sequences, we exhibit several new threshold phenomena for the order of the largest component in terms of both sources of randomness. We also provide an example of a degree sequence for which the order of the largest component undergoes an unbounded number of jumps in terms of the percolation parameter, giving rise to a behavior that cannot be observed without percolation.

While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits such a tree-decomposition, in fact a canonical one. Our proof exhibits a thick end at any obstruction to the existence of such tree-decompositions and builds on new methods for the analysis of the limit behaviour of strictly increasing sequences of separations.

Aldous and Fill (2002) conjectured that the maximum relaxation time for the random walk on a connected regular graph with n vertices is $(1+o(1\left)\right)\frac{3n^2}{2{\pi }^2}$. A conjecture by Guiduli and Mohar (1996) predicts the structure of graphs whose algebraic connectivity μ is the smallest among all connected graphs whose minimum degree δ is a given d. We prove that this conjecture implies the Aldous–Fill conjecture for odd d. We pose another conjecture on the structure of d-regular graphs with minimum μ, and show that this also implies the Aldous–Fill conjecture for even d. In the literature, it has been noted empirically that graphs with small μ tend to have a large diameter. In this regard, Guiduli (1996) asked if the cubic graphs with maximum diameter have algebraic connectivity smaller than all others. Motivated by these, we investigate the interplay between the graphs with maximum diameter and those with minimum algebraic connectivity. We show that the answer to Guiduli problem in its general form, that is for d-regular graphs for every $d\ge 3$ is negative. We aim to develop an asymptotic formulation of the problem. It is proven that d-regular graphs for $d\ge 5$ as well as graphs with $\delta =d$ for $d\ge 4$ with asymptotically maximum diameter, do not necessarily exhibit the asymptotically smallest μ. We conjecture that d-regular graphs (or graphs with $\delta =d$) that have asymptotically smallest μ, should have asymptotically maximum diameter. The above results rely heavily on our understanding of the structure as well as optimal estimation of the algebraic connectivity of nearly maximum-diameter graphs, from which the Aldous–Fill conjecture for this family of graphs also follows.

We investigate natural Turán problems for mixed graphs, generalizations of graphs where edges can be either directed or undirected. We study a natural Turán density coefficient that measures how large a fraction of directed edges an F-free mixed graph can have; we establish an analogue of the Erdős-Stone-Simonovits theorem and give a variational characterization of the Turán density coefficient of any mixed graph (along with an associated extremal F-free family).

This characterization enables us to highlight an important divergence between classical extremal numbers and the Turán density coefficient. We show that Turán density coefficients can be irrational, but are always algebraic; for every positive integer k, we construct a family of mixed graphs whose Turán density coefficient has algebraic degree k.

Aharoni and Ziv conjectured that if M and N are finitary matroids on E, then a certain “Hall-like” condition is sufficient to guarantee the existence of an M-independent spanning set of N. We show that their condition ensures that every finite subset of E is N-spanned by an M-independent set.