arXiv subjects
Publications and source records attributed to Natasha Morrison.
We determine the asymptotically best possible minimum vertex degree condition forcing a two-coloured $3$-graph to contain a colour-biased tight Hamilton cycle. This confirms a conjecture of H\`an, Lang, Marciano, Pavez-Sign\'e, Sanhueza-Matamala, Treglown and Z\'arate-Guer\'en.
Let $k \ge 2$ and let $\bf G = \{G_1, \ldots, G_{m}\}$ be a collection of graphs on a common vertex set of cardinality $n$. We show that if each graph in $\bf G$ has minimum degree at least $(1-\frac{1}{2k} + o(1))n$, then for every edge-colouring $\chi$ of the $k$th power of a Hamilton cycle $C_n^k$ with $m$ colours, there is a copy of $C_n^k$ in $\bf G$ such that $e \in G_{\chi(e)}$ for every edge $e$ in $C_n^k$. This generalises a result of Bowtell, Morris, Pehova, and Staden, who provided asymptotically best possible minimum degree conditions for the Hamilton cycle.
A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We show that for every finite graph $H$ that is not a clique or an independent set, there always exists a countable $H$-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite $H$-free graph $G$ such that any graph $G'\ne G$ obtained by making a locally finite set of changes to $G$ contains a copy of $H$.
We establish a best-possible minimum codegree condition for the existence of a perfect tiling of a $3$-uniform hypergraph $H$ with copies of the generalised triangle $T$, which is the 3-uniform hypergraph with five vertices $a, b, c, d, e$ and three edges $abc$, $abd$, $cde$. We also give an asymptotically-optimal minimum codegree condition for the rainbow version of the problem.
The $r$-bond bootstrap percolation process on a graph $G$ begins with a set $S$ of infected edges of $G$ (all other edges are healthy). At each step, a healthy edge becomes infected if at least one of its endpoints is incident with at least $r$ infected edges (and it remains infected). If $S$ eventually infects all of $E(G)$, we say $S$ percolates. In this paper we provide recursive formulae for the minimum size of percolating sets in several large families of graphs. We utilise an algebraic method introduced by Hambardzumyan, Hatami, and Qian, and substantially extend and generalise their work.
Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. The proper rainbow saturation number $\text{sat}^*(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. In this paper we initiate a systematic study of the proper rainbow saturation number for trees. We obtain exact and asymptotic results on $\text{sat}^*(n,T)$ for several infinite families of trees. Our proofs reveal connections to the classical saturation and semi-saturation numbers.
Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. The proper rainbow saturation number $\text{sat}^*(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. In this paper we use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on $\text{sat}^*(n,H)$ for general cliques, cycles, and complete bipartite graphs. We also provide some general lower bounds on $\text{sat}^*(n,H)$ and explore several other interesting directions.
Given graphs $G, H$ and an integer $q \ge 2$, the generalized Ramsey number, denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour $G$ such that every copy of $H$ receives at least $q$ colours. In this paper, we prove that for a fixed integer $k \ge 3$, we have $r(K_n,C_k,3) = n/(k-2)+o(n)$. This generalises work of Joos and Muybayi, who proved $r(K_n,C_4,3) = n/2+o(n)$. We also provide an upper bound on $r(K_{n,n}, C_k, 3)$, which generalises a result of Joos and Mubayi that $r(K_{n,n},C_4,3) = 2n/3+o(n)$. Both of our results are in fact specific cases of more general theorems concerning families of cycles.
A subgraph of the $n$-dimensional hypercube is called 'layered' if it is a subgraph of a layer of some hypercube. In this paper we show that there exist subgraphs of the cube of arbitrarily large girth that are not layered. This answers a question of Axenovich, Martin and Winter. Perhaps surprisingly, these subgraphs may even be taken to be induced.
For an oriented graph $D$ and a set $X\subseteq V(D)$, the inversion of $X$ in $D$ is the graph obtained from $D$ by reversing the orientation of each edge that has both endpoints in $X$. Define the inversion number of $D$, denoted $\mathrm{inv}(D)$, to be the minimum number of inversions required to obtain an acyclic oriented graph from $D$. The dijoin, denoted $D_1\rightarrow D_2$, of two oriented graphs $D_1$ and $D_2$ is constructed by taking vertex-disjoint copies of $D_1$ and $D_2$ and adding all edges from $D_1$ to $D_2$. We show that $\mathrm{inv}({D_1 \rightarrow D_2}) > \mathrm{inv}(D_1)$, for any oriented graphs $D_1$ and $D_2$ such that $\mathrm{inv}(D_1) = \mathrm{inv}(D_2) \ge 1$. This resolves a question of Aubian, Havet, H\"orsch, Klingelhoefer, Nisse, Rambaud and Vermande. Our proof proceeds via a natural connection between the graph inversion number and the subgraph complementation number.
We address a problem which is a generalization of Tur\'an-type problems recently introduced by Imolay, Karl, Nagy and V\'ali. Let $F$ be a fixed graph and let $G$ be the union of $k$ edge-disjoint copies of $F$, namely $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$, where each $F_i$ is isomorphic to a fixed graph $F$ and $E(F_i)\cap E(F_j)=\emptyset$ for all $i \neq j$. We call a subgraph $H\subseteq G$ multicolored if $H$ and $F_i$ share at most one edge for all $i$. Define $\text{ex}_F(H,n)$ to be the maximum value $k$ such that there exists $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$ on $n$ vertices without a multicolored copy of $H$. We show that $\text{ex}_{C_5}(C_3,n) \le n^2/25 + 3n/25+o(n)$ and that all extremal graphs are close to a blow-up of the 5-cycle. This bound is tight up to the linear error term.
Let $H_1$ and $H_2$ be graphs. A graph $G$ has the constrained Ramsey property for $(H_1,H_2)$ if every edge-colouring of $G$ contains either a monochromatic copy of $H_1$ or a rainbow copy of $H_2$. Our main result gives a 0-statement for the constrained Ramsey property in $G(n,p)$ whenever $H_1 = K_{1,k}$ for some $k \ge 3$ and $H_2$ is not a forest. Along with previous work of Kohayakawa, Konstadinidis and Mota, this resolves the constrained Ramsey property for all non-trivial cases with the exception of $H_1 = K_{1,2}$, which is equivalent to the anti-Ramsey property for $H_2$. For a fixed graph $H$, we say that $G$ has the anti-Ramsey property for $H$ if any proper edge-colouring of $G$ contains a rainbow copy of $H$. We show that the 0-statement for the anti-Ramsey problem in $G(n,p)$ can be reduced to a (necessary) colouring statement, and use this to find the threshold for the anti-Ramsey property for some particular families of graphs.
For any fixed $d\geq1$ and subset $X$ of $\mathbb{N}^d$, let $r_X(n)$ be the maximum cardinality of a subset $A$ of $\{1,\dots,n\}^d$ which does not contain a subset of the form $\vec{b} + rX$ for $r>0$ and $\vec{b} \in \mathbb{R}^d$. Such a set $A$ is said to be \emph{$X$-free}. The Multidimensional Szemer\'edi Theorem of Furstenberg and Katznelson states that $r_X(n)=o(n^d)$. We show that, for $|X|\geq 3$ and infinitely many $n\in\mathbb{N}$, the number of $X$-free subsets of $\{1,\dots,n\}^d$ is at most $2^{O(r_X(n))}$. The proof involves using a known multidimensional extension of Behrend's construction to obtain a supersaturation theorem for copies of $X$ in dense subsets of $[n]^d$ for infinitely many values of $n$ and then applying the powerful hypergraph container lemma. Our result generalizes work of Balogh, Liu, and Sharifzadeh on $k$-AP-free sets and Kim on corner-free sets.
We show that $(n,d,\lambda)$-graphs with $\lambda=O(d/\log^3 n)$ are universal with respect to all bounded degree spanning trees. This significantly improves upon the previous best bound due to Han and Yang of the form $\lambda=d/\exp{(O(\sqrt{\log n}))}$, and makes progress towards a problem of Alon, Krivelevich, and Sudakov from 2007. Our proof relies on the existence of sorting networks of logarithmic depth, as given by a celebrated construction of Ajtai, Koml\'os and Szemer\'edi. Using this construction, we show that the classical vertex-disjoint paths problem can be solved for a set of vertices fixed in advance.
A graph $H$ is common if the limit as $n\to\infty$ of the minimum density of monochromatic labelled copies of $H$ in an edge colouring of $K_n$ with red and blue is attained by a sequence of quasirandom colourings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair $(H_1,H_2)$ of such graphs, there exists $p\in(0,1)$ such that an appropriate linear combination of red copies of $H_1$ and blue copies of $H_2$ is minimized by a quasirandom colouring in which $p\binom{n}{2}$ edges are red; such a pair $(H_1,H_2)$ is said to be $(p,1-p)$-common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a $(p,1-p)$-common pair $(H_1,H_2)$ such that $H_2$ is uncommon.
For a finite abelian group $A$, define $f(A)$ to be the minimum integer such that for every complete digraph $\Gamma$ on $f$ vertices and every map $w:E(\Gamma) \rightarrow A$, there exists a directed cycle $C$ in $\Gamma$ such that $\sum_{e \in E(C)}w(e) = 0$. The study of $f(A)$ was initiated by Alon and Krivelevich (2021). In this article, we prove that $f(\mathbb{Z}_p^k) = O(pk (\log k)^2)$, where $p$ is prime, with an improved bound of $O(k \log k)$ when $p = 2$. These bounds are tight up to a factor which is polylogarithmic in $k$.
A collection of families $(\mathcal{F}_{1}, \mathcal{F}_{2} , \cdots , \mathcal{F}_{k}) \in \mathcal{P}([n])^k$ is cross-Sperner if there is no pair $i \not= j$ for which some $F_i \in \mathcal{F}_i$ is comparable to some $F_j \in \mathcal{F}_j$. Two natural measures of the `size' of such a family are the sum $\sum_{i = 1}^k |\mathcal{F}_i|$ and the product $\prod_{i = 1}^k |\mathcal{F}_i|$. We prove new upper and lower bounds on both of these measures for general $n$ and $k \ge 2$ which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patk\'{o}s, and Sz\'{e}csi from 2011.
Given a graph $H$, we say that an edge-coloured graph $G$ is $H$-rainbow saturated if it does not contain a rainbow copy of $H$, but the addition of any non-edge in any colour creates a rainbow copy of $H$. The rainbow saturation number $\text{rsat}(n,H)$ is the minimum number of edges among all $H$-rainbow saturated edge-coloured graphs on $n$ vertices. We prove that for any non-empty graph $H$, the rainbow saturation number is linear in $n$, thus proving a conjecture of Gir\~{a}o, Lewis, and Popielarz. In addition, we also give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Gir\~{a}o, Lewis, and Popielarz.