Restricted size Ramsey number for P 3 versus cycle

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Introduction
Paul Erdős had a tremendous impact on many areas of mathematics, one of these areas is Ramsey theory.His contributions started with the classical Ramsey numbers r(G, H).In 1978 Erdős et al. in [2] defined the size Ramsey number r(G, H) as the smallest size of a graph F such that, under any 2coloring of its edges, the graph F contains a red copy of G or a blue copy of H.In [5] one can find a survey of results along with the influence of Paul Erdős on the development of size Ramsey theory.
The restricted size Ramsey number r * (G, H) is a problem connecting Ramsey number and size Ramsey number.For the restricted size Ramsey number, if r is the Ramsey number of G and H then F must be a spanning subgraph of K r with the smallest size such that for any 2-coloring of edges of F we have a red copy of G or a blue copy of H in F .Therefore, the size of K r is the upper bound for the restricted size Ramsey number of G and H and the restricted size Ramsey number must be greater or equal to the size Ramsey number for a given pair of graphs.In addition, we have r(G, H) ≤ r(G, H), where r(G, H) is the on-line Ramsey number.If both G and H are complete graphs then F = K r (see [2]).The case of complete graph is one of a few cases for which that upper bound is reached.In general, the more sparse both graphs G and H are, the problem of finding the restricted size Ramsey number for those pair of graphs is harder.Only two results for the exact value of restricted size Ramsey number involving a class of graph known so far, that are, for K 1,k versus K n [6] and G versus K 1,k , where G is K 3 , K 4 −e, or C 5 [3].For other few classes of graphs, the problem is solved partially.
For notation and graph theory terminology we in general follow [8].

Known results
In this section, we list a few known definitions and theorems that we will need in proving our results.The Turán number ex(n, G) is the maximum number of edges in any nvertex graph which does not contain a subgraph isomorphic to G. A graph on n vertices is said to be extremal with respect to G if it does not contain a subgraph isomorphic to G and has exactly ex(n, G) edges.
In 1989 Clapham et al. [1] determined all values of ex(n, C 4 ) for n ≤ 21.They also characterized all the corresponding extremal graphs.In Theorem 1 we quote value for ex(7, C 4 ) and we show (Figure 1) five corresponding extremal graphs.We will use this in the proof of Theorem 5.
In our work we will also use a well known the Ramsey number for paths and cycles that was calculated by Faudree et al. in [4].

Theorem 2 ([4]
) For all integers n ≥ 4, In 2015, Silaban et al. [8] proved the lower and the upper bound for the restricted size Ramsey number for P 3 and cycles.At the end of our article we improve the upper bound for this number.

New results
In order to find the value of r * (P 3 , C n ), we must find a graph F with the smallest possible size such that F → (P 3 , C n ).According to Theorem 2 the graph F must have n vertices.
3.1 Determining the value of r * (P 3 , C 7 ) First, we give the following condition for graph F satisfying F → (P 3 , C 7 ).
Let F be a graph on 7 vertices and 12 edges.By Lemma 4, if F → (P 3 , C 7 ), then C 4 F and therefore F is one of the five graphs G i , 1 ≤ i ≤ 5 from Figure 1.Furthermore, since ∆(G i ) ≥ 4 for i ∈ {1, 2, 3}, and by coloring u 2 u 7 , u 3 u 5 , v 1 v 6 , v 3 v 7 , v 4 v 5 in red (see Figure 2) we obtain, for all G i , a 2-coloring of edges which contains neither a red P 3 nor a blue C 7 .In fact, if ∆(G i ) ≥ 4, then there is a vertex of degree at most 2 in G i .To avoid a blue C 7 we color in red one edge coming out of this vertex (if any).Hence, F (P 3 , C 7 ) and consequently we have r * (P 3 , C 7 ) ≥ 13.Next, we will show that r * (P 3 , C 7 ) ≤ 13.Let F 7 be the complement of the graph shown in Figure 3.To prove that F 7 → (P 3 , C 7 ), let χ be any 2-coloring of edges of F 7 such that there is no red P 3 in F 7 .We will show that the coloring χ will imply a blue C 7 in F 7 .To do so, consider vertex v 4 .There are 4 edges incidence to this vertex, at most one of them can be colored by red.Up to the symmetry of F 7 , without loss of generality, we can assume that v 1 v 4 is red or all edges v i v 4 , i ∈ {1, 3, 6, 7} are blue.Nonexistence a red P 3 forces the red edges to be a matching and that it suffices to consider Figure 2: Two extremal graphs G 4 and G 5 for ex(7, C 4 ).maximum matchings.Then, using symmetries, there are only five subcases to discuss.
2. All edges v i v 4 , i ∈ {1, 3, 6, 7} are blue.Then we have two subcases: 2.1 if v 2 v 5 , v 1 v 6 , v 3 v 7 are red, then we obtain the following blue cycle: For all cases, there is always a blue C 7 , so F 7 → (P 3 , C 7 ) and the proof is complete.✷ 3.2 Upper bounds for r * (P 3 , C n ) In [8] Silaban et al. proved that r * (P 3 , C n ) ≤ 2n − 1.In this section we will show that this upper bound can be improved and we prove the following theorem.
Theorem 6 For n ≥ 12, n is even, Proof.Let t = n−2 2 and let F n be a graph with where (see Fig. 4).In order to prove that F n → (P 3 , C n ), let χ be any 2-coloring of edges of F n such that there is no red P 3 in F n .We will show that the coloring F n will imply a blue C n in F n .
FACT 1. Observe that if we have any two independent blue paths to u i and v i , then we can extend these paths step by step to vertices u j and to v j for 1 ≤ i < j ≤ t.To do so, let us consider the vertex u i .Since under the coloring χ there is no red P 3 , at most one of edges {u i u i+1 , u i v i+1 } can be red.If u i u i+1 is red, then {u i v i+1 , v i u i+1 } must be blue.Using these 2 blue edges, we can extend our blue paths to u i+1 and v i+1 , independently.If u i v i+1 is red, then {u i u i+1 , v i v i+1 } must be blue.Using these 2 blue edges, we also can extend our blue paths to u i+1 and v i+1 , independently.We can do the same process to extend our blue paths until reaching u j and v j .
FACT 2. There are always two independent blue paths from x to u i and from x to v i for i = 1 or i = 3.To prove this fact, let us consider the the vertex x.There are 3 incident edges to this vertex, at most one of them can be colored by red.Up to the symmetry of F n , we can assume that at most one edge of set {xu 1 , xu 3 } is red.
If xu 3 is red, then xu 1 and xv 1 must be blue, therefore we have two blue paths from x to u 1 and from x to v 1 .Note that a similar situation occurs if none of edges incidence to x is red.Now we can assume that xu 1 is red.In this case xv 1 and xv 3 are blue so we have one path from x to u 3 .We will construct a path of size 6 with the set {u 1 , u 2 , v 1 , v 2 }) as inner vertices, namely the path from x to v 3 .To do this consider the vertex u 2 .Under the coloring χ, at most one of edges {u 2 u 3 , u 2 v 3 , u 2 v 1 } can be red.In all cases we obtain one among two possible blue paths from Similarly, using the symmetry of F n , we get two independent blue paths from y to u j and from y to v j for j = t or j = t − 2.
By using Fact 1 and 2, we obtain a blue cycle C n in F .Observe that the theorem holds for 3 ≤ t − 2 and n ≥ 12. ✷ Silaban et al. [8] gave the upper bound for the restricted size Ramsey number of P 3 versus P n .They proved that for even n > 8, r * (P 3 , P n ) ≤ 2n−1.From the proof of Theorem 6 we see that if we delete edge xu 3 then for any

Computational Approach
In this subsection we use a computational approach to determine the exact values of r * (P 3 , C n ), 8 ≤ n ≤ 12.We use the following Algorithm 1 to find such numbers.
Algorithm 1 Deciding whether graph F → (P 3 , C n ) or not Require: Adjacency matrix of biconnected graph F on n vertices.Ensure: for every subset S of m edges that compose independent edge set do 3: find a Hamiltonian cycle in F ′

5:
if no Hamiltonian cycle in F ′ then return F (P 3 , C n ), Break.end for 8: end for 9: return F → (P 3 , C n ) We generate all the adjacency matrices of biconnected graphs with n vertices (8 ≤ n ≤ 12) with minimum degree 3 by using a program called geng [7].
From the above algorithm, we obtain the results which are presented in Table 1.This table provides the value of r * (P 3 , C n ) and the number of non-isomorphic graphs F of order n and size r * (P 3 , C n ) such that F → (P 3 , C n ).Based on computer calculations, it turned out that the value of Examples of such graphs are presented in Fig. 5, 6, 7 and 4. For the number r * (P 3 , C 8 ) an example is a graph K 4,4 − e.

Conclusion
In this paper we established six new restricted size Ramsey numbers r * (P 3 , C n ) for 7 ≤ n ≤ 12.In addition, we gave the new upper bound for n ≥ 10 and n is even.It follows that the first open case of r * (P 3 , C n ) is now r * (P 3 , C 13 ) and is certainly worth of further investigation.Based on results known earlier and described in this work as well as computer experiments for some bipartite graphs that are not presented here, let us formulate the following conjecture.

Acknowledgment
We would like to thank the student of the University of Gdańsk Maciej Godek for the independent performance of some computer experiments that confirmed the correctness of the results contained in the article.
red and the remaining edges of F in blue, we obtain a 2-coloring of F which contains neither a red P 3 nor a blue C 7 .✷ Theorem 5 r * (P 3 , C 7 ) = 13.

Figure 3 :
Figure 3: The complement of the graph F 7 .

Table 1 :
Restricted size Ramsey numbers r * (P 3 , C n ), 8 ≤ n ≤ 12. of edges of F n \{xu 3 } that avoid red P 3 , it must imply a blue P n in F n .It means we get a better upper bound of the restricted size Ramsey number for P 3 versus P n , n ≥ 12 is even, as given in the following corollary.Corollary 7 For n ≥ 12 and n is even, r * (P 3 , P n ) ≤ 2n − 3.