Exact Values for Some Size Ramsey Numbers of Paths and Cycles

For the graphs G₁, G₂, and G, if every 2-coloring (red and blue) of the edges of G results in either a copy of blue G₁ or a copy of red G₂, we write G → (G₁, G₂). The size Ramsey number $\widehat{R}(G_1,G_2)$ is the smallest number e such that there is a graph G with size e satisfying G → (G₁, G₂), i.e., $\widehat{R}(G_1,G_2)=min\{|E(G)|:G\to (G_1,G_2)\}$. In this paper, by developing the procedure and algorithm, we determine exact values of the size Ramsey numbers of some paths and cycles. More precisely, we obtain that $\widehat{R}(C_4,C_5)=19$, $\widehat{R}(C_6,C_6)=26$, $\widehat{R}(P_4,C_5)=14$, $\widehat{R}(P_4,P_5)=10$, $\widehat{R}(P_4,P_6)=14$, $\widehat{R}(P_5,P_5)=11$, $\widehat{R}(P_3,P_5)=7$ and $\widehat{R}(P_3,P_6)=8$.

1. Introduction

We use standard notions and symbols from the field of graph theory, see [1]. By G = G(V, E), we denote a simple graph with vertex and edge sets V and E having cardinalities |V(G)| and |E(G)|, respectively. For S₁, S₂ ⊆ V(G), we denote E(S₁) = {uv ∈ E(G)|v, u ∈ S₁} and E(S₁, S₂) = {uv ∈ E(G)|u ∈ S₁, v ∈ S₂}. Moreover, we denote: the degree of a vertex v in G by d(v|G) (or d(v)), the minimum degree among the vertices of G by δ(G), a path and a cycle having i vertices by P_i and C_i, respectively. For the graphs G₁, G₂, and G, if every 2-coloring (red and blue) of the edges of G results in either a copy of blue G₁ or a copy of red G₂, we call it Ramsey property of G and write G → (G₁, G₂). The size Ramsey number $\widehat{R}(G_1,G_2)$ is the smallest number e such that there is a graph G with size e satisfying G → (G₁, G₂), i.e., $\widehat{R}(G_1,G_2)=min\{|E(G)|:G\to (G_1,G_2)\}$. For k ∈ ℕ, a non-complete graph G is called k-connected if |V(G)| > k and G − X is connected for every set X ⊆ V with |X| < k. The greatest integer k such that G is k-connected is the connectivity κ(G) of G. For the complete graph K_n, we define κ(K_n) = n − 1.

In 1978, Erdös et al. initiated the study of the size Ramsey number, and later it was continued by Faudree [2, 3], Lortz and Mengersen [4], and Pikhurko [5]. From these studies, we can see that the size Ramsey number $\widehat{R}(G_1,G_2)$ exists for the graphs G₁ and G₂. Su and Shao applied a backtracking algorithm to find some upper bounds for the size Ramsey numbers. The study of the size Ramsey numbers based on the graph coloring is implicitly connected to several branches of science, such as: the energies of the status level “fully functional nodes,” “partially functional nodes,” and “non-functional nodes” can be interpreted by the way of graph coloring [6], frequency channel assignment [7, 8], time tabling [9], and CAD problems [10, 11]. For more literature regarding the Ramsey numbers, we refer [12–16] to the readers. This paper is devoted to study the properties of the graphs G with the smallest size for which G → (G₁, G₂) for given graphs G₁ and G₂. Moreover, by developing the procedure and algorithm, we determined size Ramsey numbers of some paths and cycles.

2. The Approach

LEMMA 1. Let G be a graph with the smallest size for which G → (G₁, G₂). Then any G′, obtained by removing all the isolated vertices of G, is connected.

PROOF: By the definition of G′, we have $G^{\prime }\to (G_1,G_2)$. Suppose to the contrary that there are at least two components H₁, H₂ in G′. Let $G^{\prime }=H_1\cup H_2\cup \dots \cup H_n$ with n ≥ 2. Since H_i is not an isolated vertex for any i, we have $|E(H_i)|<|E(G^{\prime })|$ for any i. Then there is a 2-coloring (red and blue) f_i of the edges of H_i such that H_i contains neither red G₁ nor blue G₂. Now, consider a 2-edge coloring f of the edges of G′ with f(e) = f_i(e) for any e ∈ H_i for i = 1, 2, ⋯ , n. Then G contains neither red G₁ nor blue G₂ under f, and so $G^{\prime }\nrightarrow (G_1,G_2)$, a contradiction.

Remark 1: Given the graphs G, G₁, G₂ with G → (G₁, G₂), by the Lemma 1, we only need to consider the connected graphs for G.

LEMMA 2. If G is a graph with the smallest size for which G → (G₁, G₂), and G is a connected graph, then κ(G) ≥ min{κ(G₁), κ(G₂)}.

PROOF: Assume on contrary that we have κ(G) < min{κ(G₁), κ(G₂)}. Let S ⊆ V(G) such that |S| = κ(G) and G − S is disconnected and assume G − S = H₁ ∪ H₂ ∪ … ∪ H_n with n ≥ 2. Let V(T_i) = V(H_i) ∪ S and E(T_i) = E(H_i) ∪ E(H_i, S). Since G is a graph with the smallest size for which G → (G₁, G₂), there is a red-blue coloring f_i of the edges of T_i such that T_i contains neither red G₁ nor blue G₂ for any i. Let E(S) = {e₁, e₂, ⋯ , e_k} for some k. Now consider a 2-edge coloring f of the edges of G with f(e) = f_i(e) for any e ∈ H_i for i = 1, 2, ⋯ , n, f(e₁) = red, f(e_i) = blue for any i = 2, 3, ⋯ , k. Then G contains neither red G₁ nor blue G₂ under f, and so $G^{\prime }\nrightarrow (G_1,G_2)$. Now, we consider the following two cases:

Case 1: If there is a red copy of G₁ as a subgraph of G.

Subcase 1.1: E(G₁) ⊆ E(T_i) ∪ E(S) with i ∈ {1, …, n}.

Since f_i is a red-blue coloring of the edges of T_i such that T_i contains no red G₁. Then E(G₁) ∩ E(S) ≠ ∅. Since G₁[E(G₁) ∩ E(S)] is not a clique with |S| vertices, there is a cut-set S₁ of G₁ with S₁ ⊆ S. Then |S₁| ≤ |S| < κ(G₁) by the assumption, a contraction.

Subcase 1.2: E(G₁) ∩ E(H_i) ≠ ∅, E(G₁) ∩ E(H_j) ≠ ∅ with i ≠ j.

Then S is a cut-set of G₁ with |S| < κ(G₁) by the assumption, a contraction.

Case 2: If there is a blue copy of G₂ as a subgraph of G.

Subcase 2.1: E(G₂) ⊆ E(T_i) ∪ E(S) with i ∈ {1, …, n}.

Since f_i is a red-blue coloring of the edges of T_i such that T_i contains no blue G₂. Then E(G₂) ∩ E(S) ≠ ∅. Since G₂[E(G₂) ∩ E(S)] is not a clique with |S| vertices, there is a cut-set S₂ of G₂ with S₂ ⊆ S. Then |S₂| ≤ |S| < κ(G₂) by the assumption, a contraction.

Subcase 2.2: E(G₂) ∩ E(H_i) ≠ ∅, E(G₂) ∩ E(H_j) ≠ ∅ with i ≠ j.

Then S is a cut-set of G₂ with |S| < κ(G₂) by the assumption, a contraction.

LEMMA 3. For the graphs G, G₁ and G₂, if there exist vertices v₁, …, v_t for some 1 ≤ t ≤ |V(G)| satisfying that $d(v_i|G^{i-1})<\delta (G_1)+\delta (G_2)-1$ for any i = 1, 2, ⋯ , t and G_t ↛ (G₁, G₂), where $G^i=G-\{v_1,\dots ,v_i\}$ and G⁰ = G. Then G ↛ (G₁, G₂).

PROOF: We apply induction on t to prove it. Firstly, it is clear that the lemma holds if t = 1. Now, we suppose the stated result holds for t = i, we need to prove it for t = i + 1. Since the lemma holds if t = i, we have $G^1\nrightarrow (G_1,G_2)$. Then there is a red-blue coloring g of the edges of G¹ such that there is neither a red copy of G₁ nor a blue copy of G₂ in G¹. Let E(w) = {uv ∈ E(G)|u = w or v = w}. Since $d(v_1|G^0)<\delta (G_1)+\delta (G_2)-1$, we can divide E(v₁) into E₁, E₂ with |E₁| < δ(G₁), |E₂| < δ(G₂). Let f be a coloring of G obtained by assigning red to E₁, blue to E₁ based on g.

Case 1: If there is a red copy of G₁ as a subgraph of G under f, then v₁ ∈ V(G₁). Since |E₁| < δ(G₁), then d(v₁|G₁) < δ(G₁), a contraction.

Case 2: If there is a blue copy of G₂ as a subgraph of G under f, then v₁ ∈ V(G₂). Since |E₂| < δ(G₂), then d(v₁|G₂) < δ(G₂), a contraction.

There is neither a red copy of G₁ nor a blue copy of G₂ in G under f. Therefore, G ↛ (G₁, G₂).

The contrapositive of the Lemma 3 for t = 1 produces the following corollary:

COROLLARY 1. For any graphs G₁ and G₂, if G is any graph with the smallest size for which G → (G₁, G₂), then δ(G) ≥ δ(G₁) + δ(G₂) − 1.

LEMMA 4. For any graphs G₁ and G₂, if G is any graph with order n and size m such that G ↛ (G₁, G₂), then for any graph G′ with order at most n and size $m-1<\frac{n(n-1)}{2}$, we have $G^{\prime }\nrightarrow (G_1,G_2)$.

PROOF: First, we have G′ is not a complete graph, then there are two vertices u, v with uv ∉ E(G′). Now, we insert the edge uv to obtain a graph G″ based on G′. Then G″ is a graph with m edges and n vertices and so $G^{″}\nrightarrow (G_1,G_2)$. Therefore, there is a red − blue coloring f of G″ such that there is neither a red copy of G₁ nor a blue copy of G₂ in G″ under f. Then, there is also neither a red copy of G₁ nor a blue copy of G₂ in G′ under $f{|}_{G^{\prime }}$. Then $G^{\prime }\nrightarrow (G_1,G_2)$.

By applying the Lemma 1 and the Corollary 1, we only need to consider the connected graphs, and then propose the following algorithm (FindSizeRamseynumber) to find the size Ramsey number of G₁ and G₂. We will use the software nauty [17] to generate non-isomorphic graphs with necessary properties. If G₁ and G₂ are k-connected graphs, we further apply the Lemma 2 to reduces the number of graphs needed to be processed. For testing if G → (G₁, G₂), we applying the backtracking procedure proposed in [18].

Procedure Find(m,n,G₁,G₂);

input: m, n be integers;

        graphs G₁ and G₂.

begin

    generate the family ${G}$ of all the non-isomorphic connected graphs with size m and

    order n with minimum degree δ(G₁) + δ(G₂) − 1; (Apply Lemma 1 and Corollary 1);

    foreach G in ${G}$

    if (G → (G₁, G₂))

        return true;

    end if

    end for

     return true;

end.

Algorithm FindSizeRamseynumber(G₁,G₂);

input: graphs G₁ and G₂.

begin

1 :   Find a graph G such G → (G₁, G₂);

2 :   m = |E(G)| − 1;

3 :   $n=min\{\lfloor \frac{2m}{\delta (G_1)+\delta (G_2)-1}\rfloor ,m+1\}$;

4 :   while Find(m, n, G₁, G₂) do;

5 :   n = n − 1;

6 :    if $m>\frac{n(n-1)}{2}$ do

7 :    m = m − 1;

8 :    $n=min\{\lfloor \frac{2m}{\delta (G_1)+\delta (G_2)-1}\rfloor ,m+1\}$;

9 :    end if

10:  end while

11:  return m + 1.

end.

3. Results

EXAMPLE 1. $\widehat{R}(C_4,C_5)=19$.

PROOF: Consider G₁ = C₄, G₂ = C₅. By Algorithm FindSizeRamseynumber, we first find the graph H satisfying H → (C₄, C₅) (line 1). Therefore, $\widehat{R}(C_4,C_5)\le 19$. Then, we consider the edge number less than 19 (i.e., m ≤ 18, by line 2), and the order of graph at most $min\{\lfloor \frac{2m}{\delta (G_1)+\delta (G_2)-1}\rfloor ,m+1\}\le 12$. Now, the procedure will check if there is no graph G with minimum degree 3, size at most and order from 7 to 12 satisfying G → (C₄, C₅) (line 3-10). In this case, by applying Procedure Find, we find that there is no such graph. Therefore, $\widehat{R}(C_4,C_5)\ge 19$.

By applying Algorithm FindSizeRamseynumber, we obtain many size Ramsey numbers presented in Table 1, where #A(n, m) denote the number of non-isomorphic connected graphs with minimum degree δ(G₁) + δ(G₂) − 1 with size m and order n, and #B(n, m) denote the number of such graphs G with G → (G₁, G₂). An application of the algorithm can be used in some other graph problems, see [19].

TABLE 1

Table 1. Exact values $\widehat{R}(G_1,G_2)$ of the size Ramsey numbers of some paths and cycles.

4. Conclusion

It is a very hard task to determine the size Ramsey number even for small graphs. Faudree and Sheehan gave a table of the size Ramsey numbers for graphs with order not more than four [3]. Su and Shao [18] provide upper bounds for the size Ramsey numbers of some paths and cycles. Until now, very limited results on the size Ramsey numbers are known. In this paper, we have developed some computational techniques to determine many of those size Ramsey numbers. There are numerous variants of the Ramsey numbers such as ordered Ramsey numbers, size Ramsey numbers and zero-sum Ramsey numbers, see [20]. It is also very difficult to compute each variant of these Ramsey numbers. In order to compute some possible Ramsey numbers, we need to obtain the structure of the graphs by studying their mathematical properties. So, the approach of this paper may be considered to compute some challenging Ramsey numbers.

Author Contributions

All authors contributed equally in completing the current work.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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