Graph burning is a round-based game or process that discretely models the spread of influence throughout a network. We introduce a generalization of graph burning which applies to hypergraphs, as well as a variant called ”lazy” hypergraph burning. Interestingly, lazily burning a graph is trivial, while lazily burning a hypergraph can be quite complicated. Moreover, the lazy burning model is a useful tool for analyzing the round-based model. One of our key results is that arbitrary hypergraphs do not satisfy a bound analogous to the one in the Burning Number Conjecture for graphs. We also obtain bounds on the burning number and lazy burning number of a hypergraph in terms of its parameters, and present several open problems in the field of (lazy) hypergraph burning.

Graph burning is a discrete process that models the spread of influence through a network using a fire as a proxy for the type of influence being spread. This process was recently extended to apply to hypergraphs in both round-based and lazy settings. We introduce a variant of hypergraph burning that uses an alternative propagation rule for how the fire spreads - if some fixed proportion of vertices are on fire in a hyperedge, then in the next round, the entire hyperedge catches fire.We obtain bounds on the burning numbers of general hypergraphs, and introduce the concept of the burning distribution, which describes how the burning numbers change as the proportion parameter ranges over $(0,1)$ . We also obtain computational results which suggest there is a strong correlation between the automorphism group order and the lazy burning number of a balanced incomplete block design.

In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph G and then k firefighters protect k vertices. On each subsequent turn, the fire spreads to the collective unburnt neighbourhood of all the burning vertices and the firefighters again protect k vertices. Once a vertex has been burnt or protected it remains that way for the rest of the game. A common objective with respect to some infinite graph G is to determine how many firefighters are necessary to stop the fire from spreading after a finite number of turns, commonly referred to as containing the fire. We introduce the concept of distance-restricted firefighting where the firefighters' movement is restricted so they can only move up to some fixed distance d per turn rather than being able to move without restriction. We establish some general properties of this new game in contrast to properties of the original game, and we investigate specific cases of the distance-restricted game on the infinite square, strong, and hexagonal grids. We conjecture that two firefighters are insufficient on the square grid when d = 2, and we pose some questions about how many firefighters are required in general when d = 1.

Graph burning is a discrete process that models the spread of influence through a network using a fire as a proxy for the type of influence being spread. This process was recently extended to hypergraphs. We introduce a variant of hypergraph burning that uses an alternative propagation rule for how the fire spreads - if some fixed proportion of vertices are on fire in a hyperedge, then in the next round the entire hyperedge catches fire. This new variant has more potential for applications than the original model, and it is similarly viable for obtaining deep theoretical results. We obtain bounds which apply to general hypergraphs, and introduce the concept of the burning distribution, which describes how the model changes as the proportion ranges over (0,1). We also obtain computational results which suggest there is a strong correlation between the automorphism group order and the lazy burning number of a balanced incomplete block design.

A graph or hypergraph is said to be vertex-transitive if its automorphism group acts transitively upon its vertices. A classic theorem of Mader asserts that every connected vertex-transitive graph is maximally edge-connected. We generalise this result to hypergraphs and show that every connected linear uniform vertex-transitive hypergraph is maximally edge-connected. We also show that if we relax either the linear or uniform conditions in this generalisation, then we can construct examples of vertex-transitive hypergraphs which are not maximally edge-connected.

A graph is n-existentially closed if, for all disjoint sets of vertices A and B with |A∪ B|=n , there is a vertex z not in A∪ B adjacent to each vertex of A and to no vertex of B. In this paper, we investigate n-existentially closed line graphs. In particular, we present necessary conditions for the existence of such graphs as well as constructions for finding infinite families of such graphs. We also prove that there are exactly five 2-existentially closed planar line graphs. We then consider the existential closure of the line graphs of hypergraphs and present constructions for 2-existentially closed line graphs of hypergraphs.

Brushing of graphs is a graph searching process in which the searching agents are called brushes. We focus on brushing directed graphs based on a new model in which the brushes can only travel in the same direction as the orientation of the arcs that they traverse. We discuss strategies to brush directed graphs as well as values and bounds for the brushing number of directed graphs. We determine the brushing number for any transitive tournament, which we use to give an upper bound for the brushing number of directed acyclic graphs in general. We also establish exact values for the brushing numbers of complete directed graphs, rooted trees, and rotational tournaments.

An ${\ell}$-cycle system ${\mathcal F}$ of a graph $\Gamma$ is a set of ${\ell}$-cycles which partition the edge set of $\Gamma$. Two such cycle systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if no two distinct cycles from ${\mathcal F}\cup {\mathcal F}'$ share more than one edge. Orthogonal cycle systems naturally arise from face $2$-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal $\ell$-cycle systems of $\Gamma$ is said to be a set of mutually orthogonal cycle systems of $\Gamma$. Let $\mu(\ell,n)$ (respectively, $\mu'(\ell,n)$) be the maximum integer $\mu$ such that there exists a set of $\mu$ mutually orthogonal (cyclic) $\ell$-cycle systems of the complete graph $K_n$. We show that if $\ell\geq 4$ is even and $n\equiv 1\pmod{2\ell}$, then $\mu'(\ell,n)$, and hence $\mu(\ell,n)$, is bounded below by a constant multiple of $n/\ell^2$. In contrast, we obtain the following upper bounds: $\mu(\ell,n)\leq n-2$; $\mu(\ell,n)\leq (n-2)(n-3)/(2(\ell-3))$ when $\ell \geq 4$; $\mu(\ell,n)\leq 1$ when $\ell>n/\sqrt{2}$; and $\mu'(\ell,n)\leq n-3$ when $n \geq 4$. We also obtain computational results for small values of $n$ and $\ell$.

An $e$-star is a complete bipartite graph $K_{1,e}$. An $e$-star system of order $n>1$, $S_e(n)$, is a partition of the edges of the complete graph $K_n$ into $e$-stars. An $e$-star system is said to be $k$-colourable if its vertex set can be partitioned into $k$ sets (called colour classes) such that no $e$-star is monochromatic. The system $S_e(n)$ is $k$-chromatic if $S_e(n)$ is $k$-colourable but is not $(k-1)$-colourable. If every $k$-colouring of an $e$-star system can be obtained from some $k$-colouring $\phi$ by a permutation of the colours, we say that the system is uniquely $k$-colourable. In this paper, we first show that for any integer $k\geq 2$, there exists a $k$-chromatic 3-star system of order $n$ for all sufficiently large admissible $n$. Next, we generalize this result for $e$-star systems for any $e\geq 3$. We show that for all $k\geq 2$ and $e\geq 3$, there exists a $k$-chromatic $e$-star system of order $n$ for all sufficiently large $n$ such that $n\equiv 0,1$ (mod $2e$). Finally, we prove that for all $k\geq 2$ and $e\geq 3$, there exists a uniquely $k$-chromatic $e$-star system of order $n$ for all sufficiently large $n$ such that $n\equiv 0,1$ (mod $2e$).

This paper takes a new approach to the modelling and visualisation of sequestered CO $$_2$$ in subsurface reservoirs and its migration pathways. We model the gas as discretised packets moving along paths within a graph or network. The movement of the packets will be governed by chip-firing. This new approach provides a new insight into the modelling of CO $$_2$$ migration in porous media with complex geological architectures. We compare and contrast this new graph theoretic approach with traditional methods and demonstrate that similar results are obtained, with the added advantage that the new methods are quick to implement and execute. In addition, the new methods are more flexible and can more accurately capture the variation in the stratification of the encasing rock.

In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that, while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts remains NP-hard when restricted to hypergraphs with maximum edge size at most 3. We also discuss the relationship between strong vertex connectivity and the minimum transversal problem for hypergraphs, showing that there are classes of hypergraphs for which one of the problems is NP-hard, while the other can be solved in polynomial time.

In this paper we consider one of the most extensively studied graph search parameters, namely the zero-forcing number of a graph. We relate the zero-forcing number to the brushing number, which as a graph parameter gained recent attention due to certain industrial applications where a network system needs to be cleaned efficiently. In particular, we prove that the zero-forcing number of the line graph is an upper bound for the brushing number by constructing a brush configuration based on a zero-forcing set for the line graph. Eroh, Kang and Yi conjectured that the zero-forcing number of a graph is no more than the zero-forcing number of its line graph, which we settle in the affirmative. Moreover we prove that the brushing number of a graph is no more than the brushing number of its line graph. All three bounds are shown to be tight.