GRAPHS WITH THE BURNING NUMBERS EQUAL THREE

The concept of burning number is inspired by the firefighting problem, which is a new measure to describe the speed of information spread. For a general non-trivial connected graph 𝐺𝐺 , its burning number 𝑏𝑏 ( 𝐺𝐺 ) ≥ 2 , and 𝑏𝑏 ( 𝐺𝐺 ) = 2 if and only if the maximum degree of 𝐺𝐺 is | 𝐺𝐺 | − 1 or | 𝐺𝐺 | − 2 . In this paper, we characterize graphs with burning number 𝑏𝑏 ( 𝐺𝐺 ) = 3 .


INTRODUCTION
Recently, a new graph process, defined as graph burning, which is motivated by contagion processes of graphs such as graph searching paradigms (Firefighter Bonato and Nowakowski (2011)) and graph bootstrap percolation Balogh et al. (2012) is proposed by Bonato et al. in Bonato et al. (2014).The purpose of graph burning is to burn all the vertices as quickly as possible.All the graphs considered in this paper are simple, undirected, and finite.
Let  be a graph.Then the graph burning process of  is a discrete time which defined as follows.
Step 1: At time 0 t = .All the vertices in this time are unburned.
Graphs With the Burning Numbers Equal Three International Journal of Research -GRANTHAALAYAH 146 Step 2: At time 1 t ≥ .One can choose a new unburned vertex v (if such a vertex is available) to burning.And the chosen vertex v is called a source of fire.
If a vertex v is burned, then it must keep burning state until the burning process is finished.
Step 3: At time  + 1.All the unburned neighbors of vertex v are burned.
Step 4: If all the vertices of  are burned, then the process ends; Otherwise, turn to Step 2.
The vertex which burned in the time t is denoted by   .If a graph  is burned in k times, then a burning sequence of G is construct by the sources of fire in each time, denoted by ( 1 ,  2 , ⋯ ,   ).The burning number () of a graph , is the length of a shortest burning sequence of G .Furthermore, the shortest burning sequence of is defined as optimal burning sequence.
It follows from the definition of () that the burning number () of a graph G also is the minimum size of the sources of fire after the whole burning process finished.It is clear that the burning number of the star graph  1, is ( 1, ) = 2and the complete graph  is (  ) = 2.
Figure 1 indicates the burning number of 5 P is ( 5 ) = 3and( 1 ,  3 ,  5 ) is one of its optimal burning sequences.As proven in Bonato et al. (2015), determining the burning number of a graph  is NP-complete, even for some special graphs such as planar, disconnected, or bipartite graphs.So, it is interesting to study the sharp bounds of the burning number for any connected graphs and characterise the extremal graph.For a general non-trivial connected graph , its burning number () ≥ 2, and in Bonato and Nowakowski (2011), the authors showed that () = 2 if and only if the In this paper, we consider some sufficient condition on maximum degree of a graph to have () = 3.First, we list some useful notations and known results.
Suppose k is a positive integer and v is a vertex of  = ((), ()), then the set Let r be a positive integer and i P denote the path with order r .A spider graph (, ) is obtained by connecting a disjoint union of paths {  } =1  , with  ≥ 3, to a vertex u .The maximum degree of the spider graph (, ) is the degree of vertex u that is equal to s .Each subgraph i u P ∪ that has same length r is called an arm of (, ).

Proposition 1.8. [1]
Let G be a graph with order at least 2. Then ( )

GRAPHS 𝑮𝑮 WITH 𝒃𝒃(𝑮𝑮) = 𝟑𝟑
In this section, we characterise graphs with burning number () = 3.First, we consider a sufficient condition on the maximum degree of graphs with burning number () = 3.
In fact, the condition on the maximum degree in Theorem 2.1 is the best possible.We can show that the condition cannot be relaxed to  − 7 ≤ () ≤  − 3. First, we introduce a spider graph F which is illustrated in Figure 3.Note that (3,3) is a subtree of F and F is a subtree of (4,3), by Proposition 1.7, we get ((3,3)) = ((4,3)) = 4. Again, by Proposition 1.4, we directly get () = 4. Based on this, the following result is clear.
Theorem 2.3.Let  is a tree with order n and maximum degree 7 n ∆ = − .If T has an induced subgraph , then () ≥ 4.
Next, we consider the necessary condition on maximum degree and bound the number of edges for ( ) 3 b G = , respectively.
Theorem 2.4.If the burning number of a connected graph G with order n is Proof.It is clear that  ≤  − 3. Further, consider () = 3, we suppose the degree of the first and the second fire source are both ∆ , then we have ( 2 −  +  + 1) + ( + 1) + 1 ≥ .This means we get  ≥ √4−11−1 2 . This completes the proof.
Remark 1. K ∆ together first to get tree 1 T and then connecting one 1-degree vertex of 1 T and one 1-degree vertex of the last star  1, with 1 K .
Theorem 2.5.If the burning number of graphs  with order n is () = 3, then

CONCLUSION
In this paper, we determine the degree condition of a graph G when ( ) 3 b G = , and then discuss the bound of the maximum degree and the number of edges in graph  when () = 3.However, the sufficient-necessary maximum degree conditions for () = 3 are not known, which need further study in future.

𝑛𝑛
() = 3 and let  has less edges as possible, then G is a forest with three components.So |()| ≥  − 3. On the other hand, consider the fact that ( ) 2 b H = if and only if the maximum degree of H is || − 1 or || − 2. If () = 3, the maximum degree of  is 3 n ∆ ≤ − .This means we can by removing at least 2 edges from each vertex of complete graph   to estimate the number of edges in G .Thus, we are removing a cycle n C from n K to let  ≤  − 3.This implies |()| ≤ The upper bound in Theorem 2.5 meets the comet graph   −   and the lower bound meets the graph  1 ∪  2 ∪  1,−4 .
The upper bound of Theorem 2.4 meets the comet graph  4,−4 and the lower bound meets the graph  with order  and 4 − 11 is the square, H obtained from  + 1 stars  1, and 1 K by identifying one 1-degree vertex of ∆ stars