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The burning number conjecture holds asymptotically
... Upper bounds on b(G) have been gradually improved over time by several authors (see e.g. [7], [9], [28], and [34]). It is known that if the burning number conjecture holds for trees, then it holds for all connected graphs; thus, several papers have focused on determining or bounding b(G) for various classes of trees, such as spiders ( [15], [18]) and caterpillars ( [23], [29]). ...
... Norin and Turcotte [34] recently showed that b(G) = (1 + o(1)) √ n for all connected n-vertex graphs; this result, together with Lemma 25, yields the following general upper bound on b g (G). ...
Motivated by the burning and cooling processes, the burning game is introduced. The game is played on a graph G by the two players (Burner and Staller) that take turns selecting vertices of G to burn; as in the burning process, burning vertices spread fire to unburned neighbors. Burner aims to burn all vertices of G as quickly as possible, while Staller wants the process to last as long as possible. If both players play optimally, then the number of time steps needed to burn the whole graph G is the game burning number if Burner makes the first move, and the Staller-start game burning number if Staller starts. In this paper, basic bounds on are given and Continuation Principle is established. Graphs with small game burning numbers are characterized and Nordhaus-Gaddum type results are obtained. An analogue of the burning number conjecture for the burning game is considered and graph products are studied.
... Thus, to prove the conjecture in general, it suffices to prove the conjecture for trees. See [2,4,13] for asymptotic bounds on the burning number. The conjecture has been proven to be true for paths [6], spiders [7,8], trees whose non-leaf vertices have degree at least 4 [14], trees whose non-leaf vertices have degree at least 3 (on at least 81 vertices) [14], 1-caterpillars [10,12], and 2-caterpillars [10]. ...
Graph burning models the spread of information or contagion in a graph. At each time step, two events occur: neighbours of already burned vertices become burned, and a new vertex is chosen to be burned. The big conjecture is known as the {\it burning number conjecture}: for any connected graph on n vertices, all n vertices can be burned after at most time steps. It is well-known that to prove the conjecture, it suffices to prove it for trees. We prove the conjecture for sufficiently large p-caterpillars.
... Many classes of graphs have been verified to be well-burnable, including hamiltonian graphs 2 Ta Sheng Tan and Wen Chean Teh (Bonato et al., 2016), spiders (Bonato and Lidbetter, 2019;Das et al., 2018), and caterpillars (Hiller et al., 2021;Liu et al., 2020). Recently, the burning number conjecture was shown to hold asymptotically by Norin and Turcotte (2024). ...
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 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.
The burning number b(G) of a graph G was introduced by Bonato, Janssen, and Roshanbin [Lecture Notes in Computer Science 8882 (2014)] for measuring the speed of the spread of contagion in a graph. They proved for any connected graph G of order n, , and conjectured that . In this paper, we proved , which is roughly . We also settled the following conjecture of Bonato-Janssen-Roshanbin: provided both G and are connected.
We introduce a new graph parameter called the burning number, inspired by
contact processes on graphs such as graph bootstrap percolation, and graph
searching paradigms such as Firefighter. The burning number measures the speed
of the spread of contagion in a graph; the lower the burning number, the faster
the contagion spreads. We provide a number of properties of the burning number,
including characterizations and bounds. The burning number is computed for
several graph classes, and is derived for the graphs generated by the Iterated
Local Transitivity model for social networks.
The Burning Number Conjecture claims that for every connected graph G of order n, its burning number satisfies While the conjecture remains open, we prove that it is asymptotically true when the order of the graph is much larger than its growth, which is the maximal distance of a vertex to a well-chosen path in the graph. We prove that the conjecture for graphs of bounded growth reduces to a finite number of cases. We provide the best-known bound on the burning number of a connected graph G of order n, given by improving on the previously known bound. Using the improved upper bound, we show that the conjecture almost holds for all graphs with minimum degree at least 3 and holds for all large enough graphs with minimum degree at least 4. The previous best-known result was for graphs with minimum degree 23.
The burning number is a recently introduced graph parameter indicating the spreading speed of content in a graph through its edges. While the conjectured upper bound on the necessary number of time steps until all vertices are reached is proven for some specific graph classes, it remains open for trees in general. We present two different proofs for ordinary caterpillars and prove the conjecture for a generalised version of caterpillars and for trees with a sufficient number of legs. Furthermore, determining the burning number for spider graphs, trees with maximum degree three and path-forests is known to be NP-complete; however, we show that the complexity is already inherent in caterpillars with maximum degree three.
Graph burning is a deterministic discrete time graph process that can be interpreted as a model for the spread of influence in social networks. The burning number b(G) of a graph G is the minimum number of steps in a graph burning process for G. Bonato et al. (2014) conjectured that b(G)≤⌈n⌉ for any connected graph G of order n. In this paper, we confirm this conjecture for caterpillars. We also determine the burning numbers of caterpillars with at most two stems and a subclass of the class of caterpillars all of whose spine vertices are stems.
Motivated by a graph theoretic process intended to measure the speed of the spread of contagion in a graph, Bonato et al. (Burning a Graph as a Model of Social Contagion, Lecture Notes in Computer Science 8882 (2014) 13-22) define the burning number b(G) of a graph G as the smallest integer k for which there are vertices x1,…,xk such that for every vertex u of G, there is some i∈{1,…,k} with distG(u,xi)≤k−i, and distG(xi,xj)≥j−i for every i,j∈{1,…,k}.
For a connected graph G of order n, they prove that b(G)≤2n−1, and conjecture b(G)≤n. We show that b(G)≤3219⋅n1−ϵ+2719ϵ and b(G)≤12n7+3≈1.309n+3 for every connected graph G of order n and every 0<ϵ<1. For a tree T of order n with n2 vertices of degree 2, and n≥3 vertices of degree at least 3, we show b(T)≤(n+n2)+14+12 and b(T)≤n+n≥3. Furthermore, we characterize the binary trees of depth r that have burning number r+1.
Graph burning is one model for the spread of memes and contagion in social networks. The corresponding graph parameter is the burning number of a graph G, written b(G), which measures the speed of the social contagion. While it is conjectured that the burning number of a connected graph of order n is at most , this remains open in general and in many graph families. We prove the conjectured bound for spider graphs, which are trees with exactly one vertex of degree at least 3. To prove our result for spiders, we develop new bounds on the burning number for path-forests, which in turn leads to a -approximation algorithm for computing the burning number of path-forests.
We introduce a new graph parameter called the burning number, inspired by contact processes on graphs such as graph bootstrap percolation, and graph searching paradigms such as Firefighter. The burning number measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We provide a number of properties of the burning number, including characterizations and bounds. The burning number is computed for several graph classes, and is derived for the graphs generated by the Iterated Local Transitivity model for social networks.
This graduate-level textbook provides a straight-forward and mathematically rigorous introduction to the standard theory of point processes. The author's aim is to present an account which concentrates on the essentials and which places an emphasis on conveying an intuitive understanding of the subject. As a result, it provides a clear presentation of how statistical ideas can be viewed from this perspective and particular topics covered include the theory of extreme values and sampling from finite populations. Prerequisites are that the reader has a basic grounding in the mathematical theory of probability and statistics, but otherwise the book is self-contained. It arises from courses given by the author over a number of years and includes numerous exercises ranging from simple computations to more challenging explorations of ideas from the text.
Motivated by a certain communication problem we show that for any integer n and for any sequence (a1,…,ak) of binary vectors of length n, there is a binary vector z of length n whose Hamming distance from ai is strictly bigger than k−i for all 1 ≤ i ≤ k.
Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S is bounded or bounded above. The bounds for Pr {S – ES ≥ nt} depend only on the endpoints of the ranges of the summands and the mean, or the mean and the variance of S. These results are then used to obtain analogous inequalities for certain sums of dependent random variables such as U statistics and the sum of a random sample without replacement from a finite population.
A survey of graph burning
- Bonato