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 ...

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 ...

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 ...

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 ...

Let G be a graph attaining the maximum spectral radius among all connected nonregular graphs of order n with maximum degree Δ. Let λ 1 ( G ) be the spectral radius of G. A nice conjecture due to Liu et al. (2007) [19] asserts that lim n → ∞ ⁡ n 2 ...

A graph G is called perfect if ω ( H ) = χ ( H ) for every induced subgraph H of G, where ω ( H ) is the clique number of H and χ ( H ) its chromatic number. The Weak Perfect Graph Theorem of Lovász states that a graph G is perfect if and only if ...

Let H be an h-vertex graph. The vertex arboricity a r ( H ) of H is the least integer r such that V ( H ) can be partitioned into r parts and each part induces a forest in H. We show that for sufficiently large n ∈ h N, every n-vertex graph G ...

Given a finite simple graph Γ on n vertices its complementary prism is the graph Γ Γ ¯ that is obtained from Γ and its complement Γ ¯ by adding a perfect matching where each its edge connects two copies of the same vertex in Γ and Γ ¯. It ...

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 ...

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 ...

A ( 4 , 5 )-coloring of K n is an edge-coloring of K n where every 4-clique spans at least five colors. We show that there exist ( 4 , 5 )-colorings of K n using 5 6 n + o ( n ) colors. This settles a disagreement between Erdős and Gyárfás ...

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 (...

Given a graph G, consider the family of real symmetric matrices with the property that the pattern of their nonzero off-diagonal entries corresponds to the edges of G. For the past 30 years a central problem has been to determine which spectra ...

A graph G is ( 2 , m )-linked if, for any distinct vertices a 1 , … , a m , b 1 , b 2 in G, there exist disjoint connected subgraphs A , B of G such that a 1 , … , a m ∈ V ( A ) and b 1 , b 2 ∈ V ( B ). A fundamental result in structural graph ...

We prove that for any graph G of maximum degree at most Δ, the zeros of its chromatic polynomial χ G ( x ) (in C) lie inside the disc of radius 5.94Δ centered at 0. This improves on the previously best known bound of approximately 6.91Δ.

We also ...

Let H k r denote an r-uniform hypergraph with k edges and r + 1 vertices, where k ≤ r + 1 (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Turán density are π ( H k r ) ≤ k − 2 r for all k ≥ ...

Given a graph H, let g ( n , H ) denote the smallest k for which the following holds. We can assign a k-colouring f v of the edge set of K n to each vertex v in K n with the property that for any copy T of H in K n, there is some u ∈ V ( T ) such ...

Given a tree T of order n, one can contract any edge and obtain a new tree T ⁎ of order n − 1. In 1983, Jamison made a conjecture that the mean subtree order, i.e., the average order of all subtrees, decreases at least 1 3 in contracting an edge ...

We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs H of a graph G of the neighborhood set system of H is sandwiched between Ω ( ...

A graph G is q-Ramsey for a q-tuple of graphs ( H 1 , … , H q ) if for every q-coloring of the edges of G there exists a monochromatic copy of H i in color i for some i ∈ [ q ]. Over the last few decades, researchers have investigated a number of ...