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The `odd cover number' of a complete graph is the smallest size of a family of complete bipartite graphs that covers each edge an odd number of times. For $n$ odd, Buchanan, Clifton, Culver, Nie, O'Neill, Rombach and Yin showed that the odd cover number of $K_n$ is equal to $(n+1)/2$ or $(n+3)/2$, and they conjectured that it is always $(n+1)/2$. We prove this conjecture. For $n$ even, Babai and Frankl showed that the odd cover number of $K_n$ is always at least $n/2$, and the above authors and Radhakrishnan, Sen and Vishwanathan gave some values of $n$ for which equality holds. We give some new examples. Our constructions arise from some very symmetric constructions for the corresponding problem for complete hypergraphs. Thus the odd cover number of the complete 3-graph $K_n^{(3)}$ is the smallest number of complete 3-partite 3-graphs such that each 3-set is in an odd number of them. We show that the odd cover number of $K_n^{(3)}$ is exactly $n/2$ for even $n$, and we show that for odd $n$ it is $(n-1)/2$ for infinitely many values of $n$. We also show that for $r=3$ and $r=4$ the odd cover number of $K_n^{(r)}$ is strictly less than the partition number, answering a question of Buchanan, Clifton, Culver, Nie, O'Neill, Rombach and Yin for those values of $r$., Comment: 7 pages

The Ramsey numbers $R(T_n,W_8)$ are determined for each tree graph $T_n$ of order $n\geq 7$ and maximum degree $\Delta(T_n)$ equal to either $n-4$ or $n-5$. These numbers indicate strong support for the conjecture, due to Chen, Zhang and Zhang and to Hafidh and Baskoro, that $R(T_n,W_m) = 2n-1$ for each tree graph $T_n$ of order $n\geq m-1$ with $\Delta(T_n)\leq n-m+2$ when $m\geq 4$ is even.

Graph burning is a natural discrete graph algorithm inspired by the spread of social contagion. Despite its simplicity, some open problems remain steadfastly unsolved, notably the burning number conjecture, which says that every connected graph of order $m^2$ has burning number at most $m$. Earlier, we showed that the conjecture also holds for a path forest, which is disconnected, provided each of its paths is sufficiently long. However, finding the least sufficient length for this to hold turns out to be nontrivial. In this note, we present our initial findings and conjectures that associate the problem to some naturally impossibly burnable path forests. It is noteworthy that our problem can be reformulated as a topic concerning sumset partition of integers., Comment: Accepted and published by DMTCS

One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems included cover many of the areas of combinatorial mathematics that Imre is most associated with: including extremal problems on graphs, set systems and permutations, and Ramsey theory. This is a personal selection of problems which we find intriguing and deserving of being better known. It is not intended to be systematic, or to consist of the most significant or difficult questions in any area. Rather, our main aim is to celebrate Imre and his mathematics and to hope that these problems will make him smile. We also hope this collection will be a useful resource for researchers in combinatorics and will stimulate some enjoyable collaborations and beautiful mathematics.

How many graphs on an $n$-point set can we find such that any two have connected intersection? Berger, Berkowitz, Devlin, Doppelt, Durham, Murthy and Vemuri showed that the maximum is exactly $1/2^{n-1}$ of all graphs. Our aim in this short note is to give a 'directed' version of this result; we show that a family of oriented graphs such that any two have strongly-connected intersection has size at most $1/3^n$ of all oriented graphs. We also show that a family of graphs such that any two have Hamiltonian intersection has size at most $1/2^n$ of all graphs, verifying a conjecture of the above authors., Comment: 5 pages

Let $f(x)=x^8+ax^4+b \in \mathbb{Q}[x]$ be an irreducible polynomial where $b$ is a square. We give a method that completely describes the factorization patterns of a linear resolvent of $f(x)$ using simple arithmetic conditions on $a$ and $b$. As a result, we determine the exact six possible Galois groups of $f(x)$ and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials $x^8+ax^4+1 \in \mathbb{Q}[x]$. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials $x^8+ax^6+bx^4+ax^2+1 \in \mathbb{Q}[x]$., Comment: 22 pages, preprint of article accepted to Journal of Algebra and Its Applications

The burning number of a graph can be used to measure the spreading speed of contagion in a network. The burning number conjecture is arguably the main unresolved conjecture related to this graph parameter, which can be settled by showing that every tree of order $m^2$ has burning number at most $m$. This is known to hold for many classes of trees, including spiders - trees with exactly one vertex of degree greater than two. In fact, it has been verified that certain spiders of order slightly larger than $m^2$ also have burning numbers at most $m$, a result that has then been conjectured to be true for all trees. The first focus of this paper is to verify this slightly stronger conjecture for double spiders - trees with two vertices of degrees at least three and they are adjacent. Our other focus concerns the burning numbers of path forests, a class of graphs in which their burning numbers are naturally related to that of spiders and double spiders. Here, our main result shows that a path forest of order $m^2$ with a sufficiently long shortest path has burning number exactly $m$, the smallest possible for any path forest of the same order., Comment: preprint, 19 pages, submitted for journal consideration

In 2016, Bonato, Janssen, and Roshanbin introduced graph burning as a discrete process that models the spread of social contagion. Although the burning process is a simple algorithm, the problem of determining the least number of rounds needed to completely burn a graph, called the burning number of the graph, is NP-complete even for elementary graph structures like spiders. An early conjecture that every connected graph of order square of m can be burned in at most m rounds is the main motivator of this study. Attempts to prove the conjecture have resulted in various upper bounds for the burning number and validation of the conjecture for certain elementary classes of graphs. In this work, we find a tight upper bound for the order of a spider for it to be burned within a given number of rounds. Our result shows that the tight bound depends on the structure of the spider under consideration, namely the number of arms. This strengthens the previously known results on spiders in relation to the conjecture. More importantly, this opens up potential enquiry into the connection between burning numbers and certain characteristics of graphs. Finally, a tight upper bound for the order of a path forest for it to be burned within a given number of rounds is obtained, thus completing previously known partial corresponding results., Comment: 13 pages. preprint

For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-graphs needed to partition the complete $r$-graph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. An easy construction shows that $f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$, and we write $c_r$ for the least number such that $f_r(n) \leq c_r (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$. It was known that $c_r < 1$ for each even $r \geq 4$, but this was not known for any odd value of $r$. In this short note, we prove that $c_{295}<1$. Our method also shows that $c_r \rightarrow 0$, answering another open problem.

Let $f_r(n)$ be the minimum number of complete $r$-partite $r$-graphs needed to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. Graham and Pollak showed that $f_2(n) = n-1$. An easy construction shows that $f_r(n)\le (1-o(1))\binom{n}{\lfloor r/2\rfloor}$ and it has been unknown if this upper bound is asymptotically sharp. In this paper we show that $f_r(n)\le (\frac{14}{15}+o(1))\binom{n}{r/2}$ for each even $r\ge 4$.

Let $T$ be a tile in $\mathbb{Z}^n$, meaning a finite subset of $\mathbb{Z}^n$. It may or may not tile $\mathbb{Z}^n$, in the sense of $\mathbb{Z}^n$ having a partition into copies of $T$. However, we prove that $T$ does tile $\mathbb{Z}^d$ for some $d$. This resolves a conjecture of Chalcraft., Comment: 23 pages, 19 figures; slightly updated

An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph (on $n$ vertices) be? We show that there exist oriented 3-graphs whose shortest cycle has length $\frac{n^2}{2}(1+o(1))$: this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $\frac{n^2}{3}(1+o(1))$, in complete contrast to the case of 2-tournaments., Comment: 12 pages

Gallai's colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all 3 colours. What happens for more colours: if we $k$-colour the edges of the complete graph, with each colour class connected, how many of the $\binom{k}{3}$ triples of colours must appear as triangles? In this note we show that the `obvious' conjecture, namely that there are always at least $\binom{k-1}{2}$ triples, is not correct. We determine the minimum asymptotically. This answers a question of Johnson. We also give some results about the analogous problem for hypergraphs, and we make a conjecture that we believe is the `right' generalisation of Gallai's theorem to hypergraphs., Comment: Conjecture 3.5 is removed

A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on $n$ vertices with minimum colour degree at least $k$ contains a rainbow matching of size at least $k$, provided $n\geq (17/4)k^2$. In this paper, we show that $n\geq 4k-4$ is sufficient for $k \ge 4$.

In this paper we are interested in the brush number of a graph - a concept introduced by McKeil and by Messinger, Nowakowski and Pralat. Our main aim in this paper is to determine the brush number of the two-dimensional torus. This answers a question of Bonato and Messinger. We also find the brush number of the cartesian product of a clique with a path, which is related to the Box Cleaning Conjecture of Bonato and Messinger.

The trace of a family of sets $\mathcal{A}$ on a set $X$ is $\mathcal{A}|_X=\{A\cap X:A\in \mathcal{A}\}$. If $\mathcal{A}$ is a family of $k$-sets from an $n$-set such that for any $r$-subset $X$ the trace $\mathcal{A}|_X$ does not contain a maximal chain, then how large can $\mathcal{A}$ be? Patk\'os conjectured that, for $n$ sufficiently large, the size of $\mathcal{A}$ is at most $\binom{n-k+r-1}{r-1}$. Our aim in this paper is to prove this conjecture.

It is well known that a tournament (complete oriented graph) on $n$ vertices has at most ${1/4}\binom{n}{3}$ directed triangles, and that the constant 1/4 is best possible. Motivated by some geometric considerations, our aim in this paper is to consider some `higher order' versions of this statement. For example, if we give each 3-set from an $n$-set a cyclic ordering, then what is the greatest number of `directed 4-sets' we can have? We give an asymptotically best possible answer to this question, and give bounds in the general case when we orient each $d$-set from an $n$-set., Comment: 9 pages