A 2016 conjecture of Brewster, McGuinness, Moore, and Noel asserts that for $k \ge 3$, if a graph has chromatic number greater than $k$, then it contains at least as many cycles of length $0 \bmod k$ as the complete graph on $k+1$ vertices. Our main result confirms this in the $k=3$ case by showing every $4$-critical graph contains at least four cycles of length $0 \bmod 3$, and that $K_4$ is the unique such graph achieving the minimum. We make progress on the general conjecture as well, showing that $(k+1)$-critical graphs with minimum degree $k$ have at least as many cycles of length $0\bmod r$ as $K_{k+1}$, provided $k+1 \ne 0 \bmod r$. We also show that $K_{k+1}$ uniquely minimizes the number of cycles of length $1\bmod k$ among all $(k+1)$-critical graphs, strengthening a recent result of Moore and West and extending it to the $k=3$ case.

A 2018 conjecture of Brewster, McGuinness, Moore, and Noel asserts that for $k \ge 3$, if a graph has chromatic number greater than $k$, then it contains at least as many cycles of length $0 \bmod k$ as the complete graph on $k+1$ vertices. Our main result confirms this in the $k=3$ case by showing every $4$-critical graph contains at least $4$ cycles of length $0 \bmod 3$, and that $K_4$ is the unique such graph achieving the minimum. We make progress on the general conjecture as well, showing that $(k+1)$-critical graphs with minimum degree $k$ have at least as many cycles of length $0\bmod r$ as $K_{k+1}$, provided $k+1 \ne 0 \bmod r$. We also show that $K_{k+1}$ uniquely minimizes the number of cycles of length $1\bmod k$ among all $(k+1)$-critical graphs, strengthening a recent result of Moore and West and extending it to the $k=3$ case.

The domination polynomial of a graph is the polynomial whose coefficients count the number of dominating sets of each cardinality. A recent question asks which graphs are uniquely determined (up to isomorphism) by their domination polynomial. In this paper, we completely describe the complete r-partite graphs which are; in the bipartite case, this settles in the affirmative a conjecture of Aalipour, Akbari and Ebrahimi.

Let K_4^- denote the diamond graph, formed by removing an edge from the complete graph K_4. We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of K_4^-. We show that, with probability tending to 1 as $n \to \infty$, the final size of the graph produced is $\Theta(\sqrt{\log(n)} \cdot n^{3/2})$. Our analysis also suggests that the graph produced after i edges are added resembles the random graph, with the additional condition that the edges which do not lie on triangles form a random-looking subgraph.

We consider the following random graph process: fix an integer $\ell \ge 4$, and starting with $n$ isolatedvertices, add edges uniformly at random provided no such edge creates a copy of the cycle $C_{\ell}$. Usingthe differential equation method, we show that, with probability tending to 1 as $n \to \infty$, the finalgraph produced by this process has maximum degree $O( (n \log n)^{1/(\ell-1)})$ and, consequently, size$O(n^{\ell/(\ell-1)}(\log n)^{1/(\ell-1)})$. These results are sharp up to the hidden constants, improvingupon previous bounds due to Osthus and Taraz.

A 3-simplex is a collection of four sets A_1,...,A_4 with empty intersection such that any three of them have nonempty intersection. We show that the maximum size of a set system on n elements without a 3-simplex is $2^{n-1} + \binom{n-1}{0} + \binom{n-1}{1} + \binom{n-1}{2}$ for all $n \ge 1$, with equality only achieved by the family of sets either containing a given element or of size at most 2. This extends a result of Keevash and Mubayi, who showed the conclusion for n sufficiently large.

Let $H$ be a finite tree. We consider trees $T$ such that if the edges of $T$ are colored so that no color occurs more than $b$ times, then $T$ has a subgraph isomorphic to $H$ in which no color is repeated. We will show that if $H$ falls into a few classes of trees, including those of diameter at most $4$, then the minimum value of $e(T)$ is provided by a known construction, supporting a conjecture of Bohman, Frieze, Pikhurko and Smyth.

Classical Ramsey-theoretic questions on graphs involve coloring the edges of a graph with a fixed number of colors, say k, and determining whether a given monochromatic subgraph exists [13]. Anti-Ramsey questions, on the other hand, consider other restrictions on the colorings (for example, using each color at most b times [17]), and search for subgraphs in which no color is repeated [10]. An edge-coloring c of a graph G is b-bounded if no color occurs more than b times, and G is rainbow under c if c(e) 6= c(f) for all distinct edges e, f of G. Consider the following question, posed by Bohman, Frieze, Pikhurko and Smyth in [3]: given a (finite) tree H, what is the minimum size of a tree T such that under every b-bounded coloring of T a rainbow copy of H exists? Writing T Ã (H; b) for this condition, let