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The Ramsey Numbers for odd Complete Bipartite Graphs
Hasmawati
Department of Mathematics
Hasanuddin University (UNHAS)
Jalan Perintis Kemerdekaan KM.10 Makassar 90245
[email protected]
Abstract
The upper bound of Ramsey numbers on complete bipartite graph was shown, namely R(K1,p,Kn,m )
m+2p-2.
In this paper, we show that the Ramsey numbers R(K1,5,K2,5)=13. Furthermore, we show that R(K1,5,K2,m) = m+8
for odd m, m = 4n-7 and n  4.
Keywords : Ramsey number, Star, Complete bipartite graph
1. Introduction
Throughout this paper, all graphs are finite and simple. Let G be a graph. We write V(G) or V
for the vertex set of G and E(G) or E for the edge set of G. The graph 𝐺̅ is the complement of
G. A graph F=(V',E') is a subgraph of G if V'V(G) and E’ E(G). For S V(G), G[S]
represents the subgraph induced by S in G.
Let v be any vertex in G and S  V(G). The neighborhood NS(v) is the set of vertices in S
which are adjacent to v. Furthermore, we define NS[v]=NS(v) {v}. If S=V(G), then we use
N(v) and N[v] instead of NV(G)(v) and NV(G)[v], respectively. The degree of a vertex v in G is
denoted by dG(v). The minimum (maximum) degree of G is denoted by (G); (G). The order
of G, denoted by |G| is the number of its vertices.
Given two graphs G and H, the Ramsey number R(G,H) is the smallest positive integer k such
that for any graph F of order k the following holds: F contains G as a subgraph or 𝐹̅
contains H as a subgraph.
We denote the complete graph on n vertices by Kn. A graph G is a complete bipartite graph if
its vertices can be partitioned into two non-empty independent sets V1 and V2 such that its
edge set is formed by all edges that have one vertex in V1 and the other one in V2 . If V1 = n
and |V2|=m then the complete bipartite graph is denoted by Kn,m. A wheel Wm is the graph on
m+1 vertices that consists of a cycle Cm with one additional vertex being adjacent to all
vertices of Cm.
Some results about the Ramsey numbers for complete bipartite graphs have been known.
Burr [1] showed that R(K2,3,K2,3)=10. Parsons [7,8] showed that R(K1,7,K2,3)=13.
Additionally, Lawrence [5] showed that R(K1,15,K2,2) = 20.
Several results have been obtained for K1,p. For instance, Surahmat et al. [9] proved that
for n  3,
𝑅(𝐾1,𝑛−1 , 𝑊4 ) = {
2𝑛 + 1 if 𝑛 is even,
2𝑛 − 1 if 𝑛 is odd.
They also showed R(K1,n-1,W5) = 3n – 2 for n  3. In 2004 Chen et al. [2] generalized the
results, namely R(K1,n-1,W5) = 3n – 2 for odd m  5 and n  m-1. In [4], Korolova showed that
R(K1,n-1,Wm) = 3n – 2 for n = m, m+1, or m+2 where m  7 and is odd
In 2004 Rosyida [6] gave an upper bound on the Ramsey numbers of K1,p versus K2,m as
presented in Theorem A.
Theorem A. For p ≥ 3 and m ≥ 2, R(K1,p,K2,m ) ≤ m+2p – 2 .
Rosyida also proved the following two theorems.
Theorem B. For m,n ≥ 2, R(K1,3,Kn,m) = m + n + 2.
Theorem C. For m ≥ 2,
𝑅(𝐾1,4 , 𝐾2,𝑚 ) = {
𝑚 + 5 if 𝑚 is even,
𝑚 + 6 if 𝑚 is odd.
Recently, Hasmawati in [3] the Ramsey numbers for small complete bipartite graph were
obtained, e.g., R(K1,5,K2,2) = 8, and R(K1,5, K2,4) = 11.
In this paper we determine the Ramsey numbers R(K1,5,K2,m) for certain values of odd m as a
main result.
To obtain the main results, we need the following lemmas.
Lemma 1. Let G be a graph of order 2p + m – 5, p ≥ 5 and m ≥ 3. If ∆(G) ≤ p-2 and G
contains K3 or K2,3, then 𝐺̅ contains K2,m.
Lemma 2. Let G be a connected (p - 2)-regular graph of order 2p + m – 5, for p ≥ 4 and m ≥
3. If G contains no K3 and K2,3, then 𝐺̅ contains no K2,m.
2. Main Results
Theorem 1. R(K1,5,K2,3) = 10.
Proof. Given Consider a graph F with |F| = 9 with the vertex set VF={vi: 0 ≤ i ≤ 8 and the
edge set EF = {vi vi+1, vi vi+3} with all induced in VF and EF are taken in modulo 9. Clearly, F
is 4-regular and so contains no K1,5. It can be verified that for every u,v in 𝐹̅ ,
|𝑁𝐹̅ (𝑢) ∩ 𝑁𝐹̅ (𝑣)| ≤ 2. Hence, 𝐹̅ contains no K2,3. Thus, R(K1,5,K2,3) ≥ 10.
On the other hand, let F1 be a graph of order 10. Suppose F1 contains no K1,5. So 𝑑𝐹1 (𝑣) ≤ 4
for every vertex v in F1. Next, we show that 𝐹̅1 contain K2,3}. Let there exist u in F1 with
𝑑𝐹1 (𝑣) ≤ 3. Let w be any vertex in N(u). Since 𝑑𝐹1 (𝑣) ≤ 4 for each v in F1, |N[u] ∪ N[v]| ≥
7. If B = F1 / {N[u] ∪N[w]}, then |B| ≥ {10-7}=3. Thus, 𝐹̅1 [{u,w}∪ B] contains a K2,3.
Now, we assume that 𝑑𝐹1 (𝑣) ≤ 4 for each v in F1. Let u,w in F1 with (u,w) not in E(F1).
Write Z = N(u) ∩ N(w) and Q = F1/{ N[u] ∪N[w]}. Observe that |Q|=|Z|. If N(u) or N(w) is not
independent, then F1 contains a K3. By Lemma 1, 𝐹̅1 contains a K2,3. Suppose N(u) and N(w)
are independent sets. Then 𝐹̅1 [{u,w} ∪ Q] contains a K2,3 if |Z| ≥ 3 and 𝐹̅1 [{w} ∪N(u)]
contains a K2,3 if |Z| ≤ 2 (see Fig.1).
r
N(r
s
N(s
)
Figure 1. The illustration of proof |Z|=0.
Thus, we have R(K1,5,K2,3) ≤ 10. The proof is now complete. ∎
Theorem 2. R(K1,5,K2,5) = 13.
Proof. Given a graph F with |F| = 12 in Fig. 2.
We can see that the graph F is 4-reguler such that contains no K1,5. Futhermore, every two
vertices in F, say u,v, |Z(u,v)| ≤ 2 for (𝑢, 𝑣) ∉ 𝐸(𝐹) and Z(u,v)| = 0 for (𝑢, 𝑣)𝐸(𝐹) . Hence,
F contains no K3 and K2,3. By Lemma 2, 𝐹̅ contains no K2,5. So, F is good graph for K1,5 and
K2,5. Thus, R(K1,5,K2,5) ≥ 13.
On the other hand, by Theorem A, we have R(K1,5,K2,5) ≤ 13. ∎
Figure 2. Good graph of K1,5 and K2,5.
Theorem 3. For 𝑛 ≥ 4 and 𝑚 = 4𝑛 − 7, then 𝑅(𝐾1,5 , 𝐾2,𝑚 ) = 𝑚 + 8.
Proof.
By Theorem A, 𝑅(𝐾1,5 , 𝐾2,𝑚 ) ≤ 𝑚 + 8 𝑚 ≥ 2. Therefore, it suffices to prove that
𝑅(𝐾1,5 , 𝐾2,𝑚 ) ≥ 𝑚 + 8 𝑚 = 4𝑛 − 7 n≥ 4. Now, we construct a graph F on 4n vertices as
follows. Divide the vertices of F into two disjoint sets:
A={ui : 0 ≤ i ≤ 2n - 1} and B = {vi : 0 ≤ i ≤ 2n - 1}, n ≥ 4.
Define
EF = {uivi, uivi+2, uivi+5, uivi+9} : 0 ≤ i ≤ 2n - 1}
with all induced in VF and EF are taken in modulo 2n.
̅̅̅̅,
Clearly, F is a 4-regular and so contains no K1,5. We can observe that for every 𝑢, 𝑣 ∈ 𝐹
𝑁𝐹 (𝑢) ∩ 𝑁𝐹 (𝑣) = 0 if (u,v) in E(F) and |N(u) ∩ N(v)| ≤ 2 if (u,v) not in E(F). Hence, F
contains no K3 and K2,3. By Lemma 2, 𝐹̅ contains no K2,m. Hence, R(K1,5,K2,m) ≤ 4n+1 = m+8
for m = 4n - 7, n ≥4.
Thus, we have R(K1,5,K2,m) = m+8 for m = 4n – 7 and n ≥ 4. The proof is now complete. ∎
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