Tom Bohman
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View article: A note on Two-Point Concentration of the Independence Number of $G_{n,m}$
A note on Two-Point Concentration of the Independence Number of $G_{n,m}$ Open
We show that the independence number of $ G_{n,m}$ is concentrated on two values for $ n^{5/4+ ε} < m \le \binom{n}{2}$. This result establishes a distinction between $G_{n,m}$ and $G_{n,p}$ with $p = m/ \binom{n}{2}$ in the regime $ n^{5/…
View article: Two-Point Concentration of the Domination Number of Random Graphs
Two-Point Concentration of the Domination Number of Random Graphs Open
We show that the domination number of the binomial random graph G_{n,p} with edge-probability p is concentrated on two values for p \ge n^{-2/3+\eps}, and not concentrated on two values for general p \le n^{-2/3}. This refutes a conjecture…
View article: A critical probability for biclique partition of G,
A critical probability for biclique partition of G, Open
The biclique partition number of a graph G=(V,E), denoted bp(G), is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that bp(G)≤n−α(G), …
View article: Two-Point Concentration of the Independence Number of the Random Graph
Two-Point Concentration of the Independence Number of the Random Graph Open
We show that the independence number of $ G_{n,p}$ is concentrated on two values if $ n^{-2/3+ \epsilon } < p \le 1$ . This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in gen…
View article: Two-Point Concentration of the Independence Number of the Random Graph
Two-Point Concentration of the Independence Number of the Random Graph Open
We show that the independence number of $ G_{n,p}$ is concentrated on two values if $ n^{-2/3+ ε} < p \le 1$. This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in general, con…
View article: A Critical Probability for Biclique Partition of $G_{n,p}$
A Critical Probability for Biclique Partition of $G_{n,p}$ Open
The biclique partition number of a graph $G= (V,E)$, denoted $bp(G)$, is the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that $ bp(…
View article: Coprime mappings and lonely runners
Coprime mappings and lonely runners Open
For x real, let { x } $ \lbrace x \rbrace$ be the fractional part of x (that is, { x } = x − ⌊ x ⌋ $\lbrace x\rbrace = x - \lfloor x \rfloor$ ). The lonely runner conjecture can be stated as follows: for any n positive integers v 1 < v 2 <…
View article: Complexes of nearly maximum diameter
Complexes of nearly maximum diameter Open
The diameter of a strongly connected $d$-dimensional simplicial complex is the diameter of its dual graph. We provide a probabilistic proof of the existence of $d$-dimensional simplicial complexes with diameter $ (\frac{1}{d \cdot d!} - (\…
View article: Independent sets in hypergraphs omitting an intersection
Independent sets in hypergraphs omitting an intersection Open
A k ‐uniform hypergraph with n vertices is an ‐omitting system if it has no two edges with intersection size . If in addition it has no two edges with intersection size greater than , then it is an ‐system. Rödl and Šiňajová proved a sharp…
View article: Coprime Mappings and Lonely Runners
Coprime Mappings and Lonely Runners Open
For $x$ real, let $ \{ x \}$ be the fractional part of $x$ (i.e. $\{x\} = x - \lfloor x \rfloor $). The lonely runner conjecture can be stated as follows: for any $n$ positive integers $ v_1 < v_2 < \dots < v_n $ there exists a real number…
View article: A Construction for Boolean cube Ramsey numbers
A Construction for Boolean cube Ramsey numbers Open
Let $Q_n$ be the poset that consists of all subsets of a fixed $n$-element set, ordered by set inclusion. The poset cube Ramsey number $R(Q_n,Q_n)$ is defined as the least $m$ such that any 2-coloring of the elements of $Q_m$ admits a mono…
View article: A Construction for Cube Ramsey
A Construction for Cube Ramsey Open
The (poset) cube Ramsey number $R(Q_n,Q_n)$ is defined as the least~$m$ such that any 2-coloring of the $m$-dimensional cube $Q_m$ admits a monochromatic copy of $Q_n$. The trivial lower bound $R(Q_n,Q_n)\ge 2n$ was improved by Cox and Sto…
View article: Dynamic concentration of the triangle‐free process
Dynamic concentration of the triangle‐free process Open
The triangle‐free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal trian…
View article: On multicolor Ramsey numbers of triple system paths of length 3
On multicolor Ramsey numbers of triple system paths of length 3 Open
Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $ r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $ \binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $ \mathc…
View article: A natural barrier in random greedy hypergraph matching
A natural barrier in random greedy hypergraph matching Open
Let r ⩾ 2 be a fixed constant and let $ {\cal H} $ be an r -uniform, D -regular hypergraph on N vertices. Assume further that D → ∞ as N → ∞ and that degrees of pairs of vertices in $ {\cal H} $ are at most L where L = D/ ( log N ) ω (1) .…
View article: Independence Number of Graphs with a Prescribed Number of Cliques
Independence Number of Graphs with a Prescribed Number of Cliques Open
We consider the following problem posed by Erdős in 1962. Suppose that $G$ is an $n$-vertex graph where the number of $s$-cliques in $G$ is $t$. How small can the independence number of $G$ be? Our main result suggests that for fixed $s$, …
View article: A greedy algorithm for finding a large 2‐matching on a random cubic graph
A greedy algorithm for finding a large 2‐matching on a random cubic graph Open
A 2‐matching of a graph G is a spanning subgraph with maximum degree two. The size of a 2‐matching U is the number of edges in U and this is at least where n is the number of vertices of G and κ denotes the number of components. In this ar…
View article: Issue Information
Issue Information Open
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs.The scope of the journal also includes …
View article: A note on the random greedy independent set algorithm
A note on the random greedy independent set algorithm Open
Let r be a fixed constant and let H be an r-uniform, D-regular hypergraph on N vertices. Assume further that D > N^\epsilon for some \epsilon>0. Consider the random greedy algorithm for forming an independent set in H. An independent set i…