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The union vertex-distinguishing chromatic index $\chi'_\cup(G)$ of a graph $G$ is the smallest natural number $k$ such that the edges of $G$ can be assigned nonempty subsets of $[k]$ so that the union of the subsets assigned to the edges incident to each vertex is different. We prove that $\chi'_\cup(G) \in \left\{ \left\lceil \log_2\left(n +1\right) \right\rceil, \left\lceil \log_2\left(n +1\right) \right\rceil+1 \right\}$ for a graph $G$ on $n$ vertices without a component of order at most two. This answers a question posed by Bousquet, Dailly, Duch\^{e}ne, Kheddouci and Parreau, and independently by Chartrand, Hallas and Zhang., Comment: 8 pages, submitted

A graph is a $k$-threshold graph with thresholds $\theta_1, \theta_2, \dots, \theta_k$ if we can assign a real number $r_v$ to each vertex $v$ such that for any two distinct vertices $u$ and $v$, $uv$ is an edge if and only if the number of thresholds not exceeding $r_u+r_v$ is odd. The threshold number of a graph is the smallest $k$ for which it is a $k$-threshold graph. Multithreshold graphs were introduced by Jamison and Sprague as a generalization of classical threshold graphs. They asked for the exact threshold numbers of complete multipartite graphs. Recently, Chen and Hao solved the problem for complete multipartite graphs where each part is not too small, and they asked for the case when each part has size $3$. We determine the exact threshold numbers of $K_{3, 3, \dots, 3}$, $K_{4, 4, \dots, 4}$ and their complements $nK_3$, $nK_4$. This improves a result of Puleo., Comment: 24 pages, 2 figures, 2 tables, submitted

In the Toucher-Isolator game, introduced recently by Dowden, Kang, Mikala\v{c}ki and Stojakovi\'{c}, Toucher and Isolator alternately claim an edge from a graph such that Toucher aims to touch as many vertices as possible, while Isolator aims to isolate as many vertices as possible, where Toucher plays first. Among trees with $n$ vertices, they showed that the star is the best choice for Isolator and they asked for the most suitable tree for Toucher. Later, R\"{a}ty showed that the answer is the path with $n$ vertices. We give a simple alternative proof of this result. The method to determine where Isolator should play is by breaking down the gains and losses in each move of both players., Comment: 11 pages, 3 figures, 3 tables, submitted

In this paper, we prove that given a 2-edge-coloured complete graph $K_{4n}$ that has the same number of edges of each colour, we can always find a perfect matching with an equal number of edges of each colour. This solves a problem posed by Caro, Hansberg, Lauri, and Zarb. The problem is also independently solved by Ehard, Mohr, and Rautenbach., Comment: 12 pages, 6 figures, 3 tables, submitted

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. In 1999, Stacey and Weidl, partially resolving a conjecture of Erickson from 1994, showed that for a fixed natural number $m>2$ and for all sufficiently large $k$, there is a $k$-colouring of the complete graph on $\mathbb{N}$ such that no complete infinite subgraph is exactly $m$-coloured. In the light of this result, we consider the question of how close we can come to finding an exactly $m$-coloured complete infinite subgraph. We show that for a natural number $m$ and any finite colouring of the edges of the complete graph on $\mathbb{N}$ with $m$ or more colours, there is an exactly ${\hat m}$-coloured complete infinite subgraph for some ${\hat m}$ satisfying $|m-{\hat m}|\le \sqrt{m/2} + 1/2$; this is best-possible up to the additive constant. We also obtain analogous results for this problem in the setting of $r$-uniform hypergraphs. Along the way, we also prove a recent conjecture of the second author and investigate generalisations of this conjecture to $r$-uniform hypergraphs., 12 pages, fixed misprints, Journal of Graph Theory