Burning Number: Graph Spread & Coverage

• Burning number is a graph parameter that defines the minimum rounds needed to burn every vertex in a connected graph using sequential ignitions and radius covers.
• It bridges dynamic propagation models with geometric covering techniques, employing discrete rounds, closed ball formulations, and recursive deletion strategies.
• Key results include the burning number conjecture, precise extremal bounds for trees and grids, and NP-hardness along with approximation methods in various graph classes.

The burning number is a graph parameter that measures the speed at which fire, influence, or information can spread through a finite connected graph when one new source may be ignited at each discrete round and all previously burning vertices simultaneously ignite their neighbors. If $G$ can be covered by closed balls of radii $k-1,k-2,\dots,0$ centered at suitable vertices, then $G$ burns in $k$ rounds, and the minimum such $k$ is denoted $b(G)$. Introduced by Bonato, Janssen and Roshanbin in 2016, the notion sits at the intersection of graph covering, tree structure, extremal bounds, approximation algorithms, and probabilistic spreading models (Ning et al., 3 Sep 2025, Bonato, 2020).

1. Formal model and equivalent formulations

Let $G=(V,E)$ be a finite, simple, undirected graph. In the standard burning process, all vertices are initially unburned. At round $i=1,2,\dots$, one unburned vertex $x_i$ may be chosen as a new source, and every vertex that was already burning spreads fire to all of its unburned neighbors. Once a vertex is burned, it remains burned forever. A sequence $(x_1,\dots,x_k)$ is a burning sequence of length $k-1,k-2,\dots,0$0 if all vertices are burned by round $k-1,k-2,\dots,0$1, and the burning number $k-1,k-2,\dots,0$2 is the minimum such $k-1,k-2,\dots,0$3 (Ning et al., 3 Sep 2025, Guillaume et al., 2 Jun 2026).

The process admits an equivalent covering formulation. One may write

$k-1,k-2,\dots,0$4

where $k-1,k-2,\dots,0$5 is the closed ball of radius $k-1,k-2,\dots,0$6 around $k-1,k-2,\dots,0$7. In this view, the source ignited at round $k-1,k-2,\dots,0$8 has influence radius $k-1,k-2,\dots,0$9 by the end of round $G$0. An early formulation also imposed pairwise distance constraints on the sources, but a key simplification is that this non-redundancy condition is redundant: the covering condition alone suffices to define $G$1 (Bessy et al., 2015).

This equivalence makes graph burning a radius-ordered covering problem. Much of the subject therefore alternates between dynamic language—fires, rounds, propagation—and geometric language—balls, radii, and coverings in graph distance (Bonato, 2020).

2. Conjecture, extremal examples, and general bounds

The central open statement is the Burning Number Conjecture: every connected $G$2-vertex graph $G$3 satisfies

$G$4

Paths $G$5 and cycles $G$6 satisfy $G$7, so the conjectured bound is sharp. Bonato–Janssen–Roshanbin also showed that

$G$8

so proving the conjecture for trees is sufficient for all connected graphs (Ning et al., 3 Sep 2025, Bonato, 2020).

Several general bounds are known. The radius bound gives $G$9, while the survey literature records the basic lower bound $k$0. The earliest general order bound was

$k$1

This was improved first to

$k$2

and then to the explicit bound

$k$3

for every connected graph of order $k$4 (Bessy et al., 2015, Land et al., 2016, Bonato, 2020).

A major recent development is the asymptotic resolution of the conjecture: Norin and Turcotte proved that every connected $k$5-vertex graph satisfies

$k$6

Their proof passes to metric trees, develops two-ball covers for minimal subtrees, constructs randomized almost-uniform covers, and then rounds continuous radii back to integer burning schedules. This establishes the correct leading constant $k$7, while leaving the exact $k$8 statement open (Norin et al., 2022).

3. Trees and the structural frontier

Because of the spanning-tree reduction, progress on trees has been decisive. One prominent line concerns caterpillar-like trees. For an ordinary caterpillar $k$9 on $k$0 vertices,

$k$1

The same paper proves the conjectured bound for all $k$2-caterpillars and for any tree with at least $k$3 leaves. Its proof techniques include an infinite-descent argument and an explicit recursive deletion strategy that repeatedly removes burning circles of lengths $k$4, $k$5, and $k$6 so that the remaining order drops to a smaller square (Hiller et al., 2019).

A second structural line tracks degree-2 vertices. If $k$7 is a tree of order $k$8 with $k$9 vertices of degree $b(G)$0, Ning–Jin–Zhang proved

$b(G)$1

As a corollary, the conjectured bound $b(G)$2 holds whenever $b(G)$3. Their method combines a separator lemma for trees without degree-2 vertices with a smoothing operation and then restores degree-2 vertices by attaching leaves to form a supertree with no degree-2 vertices (Ning et al., 3 Sep 2025).

Algorithmic burning constructions have also been developed for binary-tree families. Perfect binary trees of height $b(G)$4 satisfy $b(G)$5; complete binary trees have burning number $b(G)$6 or $b(G)$7 depending on the last level; and full, nonperfect binary trees satisfy $b(G)$8. More generally, for a tree $b(G)$9 with $G=(V,E)$0 degree-2 vertices,

$G=(V,E)$1

improving the earlier $G=(V,E)$2-based bound for $G=(V,E)$3 (Das et al., 2023).

4. Exact values and special graph classes

The smallest nontrivial burning number is completely characterized. A graph $G=(V,E)$4 on $G=(V,E)$5 vertices satisfies $G=(V,E)$6 if and only if $G=(V,E)$7. This degree condition yields immediate exact values for complete graphs and near-complete graphs and underlies later characterizations of graph classes with constant burning number (Guillaume et al., 2 Jun 2026).

Several hereditary classes are now understood. Every connected split graph satisfies $G=(V,E)$8, with $G=(V,E)$9 only when $i=1,2,\dots$0. Every connected $i=1,2,\dots$1-free graph satisfies $i=1,2,\dots$2, and the only graph in that class with burning number $i=1,2,\dots$3 is the disconnected graph $i=1,2,\dots$4. The same work formalizes the term well-burnable for classes satisfying $i=1,2,\dots$5; connected split graphs and connected $i=1,2,\dots$6-free graphs are well-burnable (Guillaume et al., 2 Jun 2026).

Rooted graph products provide another exact family. For the comb graph

$i=1,2,\dots$7

obtained by attaching a path $i=1,2,\dots$8 to each vertex of a spine $i=1,2,\dots$9, the exact formula in the spine-dominant regime is

$x_i$0

In the tooth-dominant regime $x_i$1, one has

$x_i$2

so the Burning Number Conjecture holds for all comb graphs (Peca-Medlin, 27 Feb 2026).

For Cartesian products of paths, asymptotic behavior depends on the regime. For the grid $x_i$3,

$x_i$4

whereas $x_i$5 when $x_i$6. In the fence regime $x_i$7, explicit lower and upper bounds sharpen this $x_i$8 statement, including the lower bound

$x_i$9

and the upper bound

$(x_1,\dots,x_k)$0

(Mitsche et al., 2015, Bonato et al., 2019).

5. Computational complexity and approximation

The decision problem—given $(x_1,\dots,x_k)$1 and $(x_1,\dots,x_k)$2, decide whether $(x_1,\dots,x_k)$3—is computationally hard on highly restricted classes. It is NP-complete for trees of maximum degree $(x_1,\dots,x_k)$4, spider graphs, and path-forests. The hardness already persists for caterpillars of maximum degree $(x_1,\dots,x_k)$5, showing that the complexity is inherent even in very simple tree families. Later work strengthened this further to connected cubic graphs and connected proper interval graphs, and BNP is APX-hard on connected cubic graphs as well as on connected $(x_1,\dots,x_k)$6-regular graphs for every fixed $(x_1,\dots,x_k)$7 (Bonato et al., 2015, Hiller et al., 2019, Antony et al., 2024, Antony et al., 14 May 2026).

Approximation algorithms have therefore become central. Earlier results gave a $(x_1,\dots,x_k)$8-approximation for general graphs and a $(x_1,\dots,x_k)$9-approximation for trees. Martinsson improved the general approximation ratio to

$k-1,k-2,\dots,0$00

in polynomial time and proved a PTAS for trees and forests. The same paper shows that, for every $k-1,k-2,\dots,0$01, computing a $k-1,k-2,\dots,0$02-approximation for general graphs is NP-hard (Martinsson, 2023).

Despite these hardness results, some classes admit efficient exact computation. Spider graphs and path-forests are polynomial-time solvable when the number of arms or components is fixed. Connected split graphs admit linear-time computation of $k-1,k-2,\dots,0$03. For connected $k-1,k-2,\dots,0$04-free graphs,

$k-1,k-2,\dots,0$05

and for fixed $k-1,k-2,\dots,0$06 an optimal burning sequence can be found in

$k-1,k-2,\dots,0$07

time (Bonato et al., 2015, Guillaume et al., 2 Jun 2026, Antony et al., 2024).

6. Probabilistic, geometric, and extended models

Probabilistic graph burning studies either random input graphs or random source selection. For $k-1,k-2,\dots,0$08 with average degree $k-1,k-2,\dots,0$09, the burning number is asymptotically tied to the graph diameter, and one has a.a.s. $k-1,k-2,\dots,0$10. For random geometric graphs $k-1,k-2,\dots,0$11 above the connectivity threshold, $k-1,k-2,\dots,0$12. On paths, random source-selection models exhibit sharply different behavior: $k-1,k-2,\dots,0$13 and $k-1,k-2,\dots,0$14 are asymptotically $k-1,k-2,\dots,0$15, while $k-1,k-2,\dots,0$16 (Mitsche et al., 2015).

A different random model chooses each new ignition uniformly from the unburned vertices of the discrete torus $k-1,k-2,\dots,0$17. In that setting, the random completion time $k-1,k-2,\dots,0$18 satisfies

$k-1,k-2,\dots,0$19

where $k-1,k-2,\dots,0$20 is the explosion time of the generalized Blasius equation

$k-1,k-2,\dots,0$21

This connects random burning to a deterministic nonlinear ODE limit (Blanc et al., 2 Sep 2025).

Geometric variants replace graphs by point sets in the plane. In point burning, ignitions must occur at input points; in anywhere burning, ignitions may occur anywhere in $k-1,k-2,\dots,0$22. Both decision problems are NP-complete, and both admit $k-1,k-2,\dots,0$23-approximation algorithms for every $k-1,k-2,\dots,0$24. Under a bound on the number of sources, anywhere burning is NP-hard to approximate within a factor of $k-1,k-2,\dots,0$25 (Keil et al., 2022).

Two further variants burn edges or both vertices and edges. Edge burning is exactly burning in the line graph $k-1,k-2,\dots,0$26, so $k-1,k-2,\dots,0$27, and for every connected $k-1,k-2,\dots,0$28,

$k-1,k-2,\dots,0$29

Total burning is burning in the total graph $k-1,k-2,\dots,0$30, so $k-1,k-2,\dots,0$31, with

$k-1,k-2,\dots,0$32

These relations show that the standard burning number sits naturally inside a broader family of incidence-based propagation parameters (Antony et al., 2024).

The principal open problem remains the exact Burning Number Conjecture. Current structural work emphasizes extensions from caterpillars to $k-1,k-2,\dots,0$33-caterpillars with $k-1,k-2,\dots,0$34, characterizations of graphs with $k-1,k-2,\dots,0$35, and burning on dynamic or temporal networks (Hiller et al., 2019, Guillaume et al., 2 Jun 2026).