Journal of Combinatorial Theory Series B最新文献

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.

A graph G is called perfect if $\omega \left(H\right)=\chi \left(H\right)$ for every induced subgraph H of G, where $\omega \left(H\right)$ is the clique number of H and $\chi \left(H\right)$ its chromatic number. The Weak Perfect Graph Theorem of Lovász states that a graph G is perfect if and only if its complement $\bar{G}$ is perfect. This does not hold for box-perfect graphs, which are the perfect graphs whose stable set polytope is box-totally dual integral.

We prove that both G and $\bar{G}$ are box-perfect if and only if ${\bar{G}}^{+}$ is box-perfect, where $G^{+}$ is obtained by adding a universal vertex to G. Consequently, $G^{+}$ is box-perfect if and only if ${\bar{G}}^{+}$ is box-perfect. As a corollary, we characterize when the complete join of two graphs is box-perfect.

In this note, we prove that finite CAT(0) cube complexes can be reconstructed from their boundary distances (computed in their 1-skeleta). This result was conjectured by Haslegrave, Scott, Tamitegama, and Tan (2023). The reconstruction of a finite cell complex from the boundary distances is the discrete version of the boundary rigidity problem, which is a classical problem from Riemannian geometry. In the proof, we use the bijection between CAT(0) cube complexes and median graphs, and corner peelings of median graphs.

In a fractional coloring, vertices of a graph are assigned measurable subsets of the real line and adjacent vertices receive disjoint subsets; the fractional chromatic number of a graph is at most k if it has a fractional coloring in which each vertex receives a subset of $[0,1]$ of measure at least $1/k$. We introduce and develop the theory of “fractional colorings with local demands” wherein each vertex “demands” a certain amount of color that is determined by local parameters such as its degree or the clique number of its neighborhood. This framework provides the natural setting in which to generalize degree-sequence type bounds on the independence number. Indeed, by Linear Programming Duality, all of the problems we study have an equivalent formulation as a problem concerning weighted independence numbers, and they often imply new bounds on the independence number.

Our results and conjectures are inspired by many of the most classical results and important open problems concerning the independence number and the chromatic number, often simultaneously. We conjecture a local strengthening of both Shearer's bound on the independence number of triangle-free graphs and the fractional relaxation of Molloy's recent bound on their chromatic number, as well as a longstanding problem of Ajtai et al. on the independence number of $K_r$-free graphs and the fractional relaxations of Reed's $\omega ,\Delta ,\chi$ Conjecture and the Total Coloring Conjecture. We prove an approximate version of the first two, and we prove “local demands” versions of Vizing's Theorem and of some χ-boundedness results.

The study of graph discrepancy problems, initiated by Erdős in the 1960s, has received renewed attention in recent years. In general, given a 2-edge-coloured graph G, one is interested in embedding a copy of a graph H in G with large discrepancy (i.e. the copy of H contains significantly more than half of its edges in one colour).

Motivated by this line of research, Gishboliner, Krivelevich and Michaeli considered an oriented version of graph discrepancy for Hamilton cycles. In particular, they conjectured the following generalisation of Dirac's theorem: if G is an oriented graph on $n\ge 3$ vertices with $\delta \left(G\right)\ge n/2$, then G contains a Hamilton cycle with at least $\delta \left(G\right)$ edges pointing forwards. In this paper, we present a full resolution to this conjecture.