Research On Burning Numbers Of Unicyclic Graphs

The spread of information and viruses on social networks,as well as the outbreak of infectious diseases in our lives,are our common combusion process.The combusion process can be regarded as the process of realizing the full coverage of information.However,the spread of viruses and infectious diseases will have a bad impact on the society.Therefore,it is necessary to study how to determine the shortest time to realize the full coverage of information.In 2014,Bonato at al.proposed the concept of graph burning.The burning process of the graph was used to reflect the combusion process,and the burning number of the graph was used to reflect the combusion velocity.Given a graph G=(V(G),E(G)),(x1,x2,…,xk)is a burning sequence of G.The burning number of G is the smallest integer k for which Nk-1[x1]?Nk-2[x2]?…?N1[Xk-1]? N0[xk]=V(G),for any 1?i<j?k,d(xi,Xj)?j-i.Bessy at al.have shown that the graph burning problem is NP-complete,even for trees with maximum degree three,or unconnected graph,like linear forests(a disjoin union of paths).So it is meaningful to study the burning number of special graph or graph class.In this paper,the burning number of several kinds of unicyclic graphs are studied.Bonato at al.determined the burning number of(Cn or Pn is[(?)],and conjectured that the upper bound on the burning number of all connected graphs are[(?)].We first proved that both t-unicyclic graphs and sunlet graphs are true for Bonato conjecture.And then use the burning numbers of linear forests with at most three components to determine the burning number of all t-unicyclic graphs for t?2,finally,we determine the burning number of three classes of sunlet graphs.