The Burning Number Problem (BNP) models the spread of information or contagion in a network through a discrete-time process on a graph. At each step, one new vertex is selected as a burning source, while fire simultaneously spreads from previously burned vertices to their neighbors. The burning number of a graph is the minimum number of steps required to burn all vertices. The decision version asks whether the burning number is at most a given integer $k$. BNP is known to be NP-complete even on restricted graph classes such as path forests. We study BNP on connected regular graphs, a natural and previously unexplored graph class. We prove that BNP is NP-complete on connected cubic graphs, and moreover APX-hard under this restriction. We further show that BNP remains APX-hard on connected $d$-regular graphs for every fixed $d \geq 4$.

翻译：燃烧数问题（Burning Number Problem, BNP）通过图上的离散时间过程来建模网络中信息或传染病的传播。每一步，选择一个新顶点作为燃烧源，同时火焰从先前已燃烧的顶点向其邻居扩散。图的燃烧数是烧毁所有顶点所需的最少步骤数。其判定形式询问燃烧数是否至多为给定整数$k$。已知即使对于受限图类（如路径森林），BNP也是NP完全的。我们研究连通正则图上的BNP，这是一个自然且先前未被探索的图类。我们证明连通三次图上的BNP是NP完全的，并且在此限制下还是APX难的。进一步，我们证明对于每个固定的$d \geq 4$，连通$d$-正则图上的BNP依然是APX难的。