The Ramsey Numbers for Small Complete Bipartite Graphs
Hasmawati
Department of Mathematics
Hasanuddin University (UNHAS)
Jalan Perintis Kemerdekaan KM.10 Makassar 90245
hasma_ba@yahoo.com
Abstract
The Ramsey numbers for a graph G versus a graph H, denoted by R(G,H) is the smallest positive integer n such that
for any graph F of order n, either F contains G as a subgraph or
𝐹̅ contains H as a subgraph. 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 for small complete bipartite graph R(K1,5,K2,2)=8, R(K1,5,K2,3)=10, and R(K1,5,K2,4)=11.
Keywords : Ramsey number, Star, Complete bipartite graph
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 [6] showed that R(K1,7,K2,3)=13. Additionally, Lawrence
[4] showed that R(K1,15,K2,2) = 20.
Several results have been obtained for K1,p. For instance, Surahmat et al. [8] proved that
for n  3,
2𝑛 + 1 if 𝑛 is even,
𝑅(𝐾1,𝑛−1 , 𝑊4 ) = {
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 [3], Korolova showed that R(K1,n1,Wm) = 3n – 2 for n = m, m+1, or m+2 where m  7 and is odd.
In 2004 Rosyida [5] 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,
𝑚 + 5 if 𝑚 is even,
𝑅(𝐾1,4 , 𝐾2,𝑚 ) = {
𝑚 + 6 if 𝑚 is odd.
In this paper we determine the Ramsey numbers R(K1,5,K2,m) for certain values of m as a main
result. Before proving this the main results we will prove the next two lemmas.
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.
Proof. If G contains a K3, then select any two vertices of K3. Call them u and v. If G contains a
K2,3, then select the two vertices of K2,3 of degree of 3, namely u and v. Since d(x) ≤ p – 2 for
every vertex x in G and p ≥ 3, |N[u] ∪ N[v]| ≤ 2 p – 5. Write B = G/N[u] ∪ N[v]. Since |G| = 2 p
+ m – 5 and |N[u] ∪ N[v]| ≤ 2 p – 5, then |B| ≥ m. Hence, 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.
Proof. Let G be a connected (p - 2)-regular graph of order 2p + m – 5, for p,m ≥ 3. If G
contains no K3 and K2,3, then for any two nonadjacent vertices u,v of G, we have |N(u) ∩ N(v)| ≤
2; and for any two adjacent vertices u and v, we have |N(u) ∩ N(v)| = 0. Consequently, |N[u] ∪
N[v]| ≥ 2p-4 for all u,v in G. Write A = G / N[u] ∪ N[v]. Since |G|=2p+m-5, then |A| ≤ m-1.
Hence, 𝐺̅ contains no K2,m. ∎
1. Main Results
Theorem 1. R(K1,5,K2,2) = 8.
Proof. Given a 4-regular graph, call F with |F| = 7 so that 𝐹̅ = C7. This graph contains no K1,5
and its complement contains no K2,2. Hence, we find R(K1,5,K2,2) ≤ 8. From Parson in [7], we
obtain R(K1,5,K2,2) ≤ 6 + √5 . Therefore, we have R(K1,5,K2,2) = 8. ∎