Stars versus stripes Ramsey numbers

0.1 MB·

by  G. R. Omidi

Stars versus stripes Ramsey numbers
G.R. Omidi,a,c G. Raeisi,b,c Z. Rahimia
aDepartmentofMathematicalSciences,IsfahanUniversityofTechnology,Isfahan,84156-83111,Iran
bDepartmentofMathematicalSciences,ShahrekordUniversity,Shahrekord,P.O.Box115,Iran
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1 cSchoolofMathematics,InstituteforResearchinFundamentalSciences(IPM),
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2 P.O.Box:19395-5746,Tehran,Iran

n
[email protected],[email protected],[email protected]
a
J

6
1
Abstract

]
O
For given simple graphs G1,G2,...,Gt, the Ramsey number R(G1,G2,...,Gt) is the smallest
C
positive integer n such that if the edges of the complete graph Kn are partitioned into t disjoint
.
h color classes giving t graphs H1,H2,...,Ht, then at least one Hi has a subgraph isomorphic
t
a to Gi. In this paper, for positive integers t1,t2,...,ts and n1,n2,...,nc the Ramsey number
m
R(St1,St2,...,Sts,n1K2,n2K2,...,ncK2)iscomputed,wherenK2denotesamatching(stripe)
[
ofsizen, i.e.,npairwisedisjointedgesandSn isastarwithnedges. Thisresultgeneralizesand

1 strengthenssignificantlyawell-knownresultofCockayneandLorimerandalsoaknownresultof
v
Gya´rfa´sandSa´rko¨zy.
1
9
1
4
0
.
1
1 Introduction
0
7
1
In this paper, we only concerned with undirected simple finite graphs and we follow [1] for
:
v
terminology and notations not defined here. For a graph G, we denote its vertex set, edge set,
i
X
minimum degree, maximum degree and complement graph by V(G), E(G), δ(G), ∆(G) and
ar G¯, respectively. If v ∈ V(G), we use degG(v) and NG(v) (or simply deg(v) and N(v)) to
denote the degree and the neighbors of v in G, respectively. Also, we use nK to denote a
2
matching (stripe) of size n, i.e., n pairwise disjoint edges and as usual, a complete graph on n
vertices,astarwithnedgesandabalancedcompletebipartitegraphon2nverticesaredenoted
by K , S and K , respectively. In addition, for disjointsubsets A and B of the vertex set of
n n n,n
agraph G, weuse[A,B]to denotethebipartitesubgraphofG withpartitesets Aand B.
If G is a graph whose edges are colored by c colors, we use Gi, 1 ≤ i ≤ c, to denote the
subgraph of G induced by the edges of the i-th color. Moreover, for a vertex v of G, we use
degi(v)and Ni(v)to denotethedegreeand theneighborsofv inGi, respectively.
1
Recall that an edge coloring of G is called proper if adjacent edges are assigned different
colors. Theminimumnumberofcolors for aproper edgecoloring ofG is called thechromatic
index of G and is denoted by χ′(G). It is well known that for a bipartite graph G, we have
χ′(G) = ∆(G), see[1].
Let G,G ,G ,...,G be given simple graphs. We write G → (G ,G ,...,G ), if the
1 2 c 1 2 c
edges of G are partitioned into c disjoint color classes giving c graphs H ,H ,...,H , then
1 2 c
at least one H has a subgraph isomorphic to G . For given simple graphs G ,G ,...,G ,
i i 1 2 c
the multicolor Ramsey number R(G ,G ,...,G ) is defined as the smallest positiveinteger n
1 2 c
such that K → (G ,G ,...,G ). The existence of such a positive integer is guaranteed by
n 1 2 c
the Ramsey’s classical result [8]. For a survey on Ramsey theory, we refer the reader to the
regularlyupdatedsurveybyRadziszowski[7].
Thereis verylittleknownaboutR(G ,G ,...,G ) forc ≥ 3,evenforveryspecialgraphs.
1 2 c
In this paper, we consider the case that G ’s are stars or stripes. The Ramsey number of stars
i
or stripes were investigated by several authors. The Ramsey number of stars is determined by
Burr and Roberts [2] and the Ramsey number for stripes was determined by Cockayne and
Lorimer[3]. In fact theyshowedthat R(n K ,n K ,...,n K ) = n + c (n −1)+1 for
1 2 2 2 c c 1 i=1 i
P
n ≥ n ≥ ··· ≥ n . In [6] Gya´rfa´s and Sa´rko¨zy determined the exact value of the Ramsey
1 2 c
number of a star versus two stripes and then they used this result to givea positiveanswer to a
conjectureofSchelpinanasymptoticsense. ItisalsoworthnotingthattheRamseynumberfor
manystarsand onestripewas determinedin[4] asfollows.
Theorem 1.1. [4]Let t ,t ,...,t bepositiveintegers,Σ = s (t −1) andn ≥ 1. Then
1 2 s i=1 i
P
1)R(S ,S ,...,S ,nK ) = 2n, ifΣ < n,
t1 t2 ts 2
2)R(S ,S ,...,S ,nK ) = Σ+n, ifΣ ≥ n, Σ iseven andsomet is even,
t1 t2 ts 2 i
3)R(S ,S ,...,S ,nK ) = Σ+n+1,otherwise.
t1 t2 ts 2
Note that, using Theorem 1.1 for n = 1, we conclude that R(S ,S ,...,S ) = Σ+1, if
t1 t2 ts
Σand at leastonet areevenand R(S ,S ,...,S ) = Σ+2,otherwise.
i t1 t2 ts
The aim of this paper is the following theorem which provides the exact value of the Ramsey
number of any number of stars versus any number of stripes. This theorem extends known
resultson theRamsey numberofstarsand stripesintheliterature.
Theorem 1.2. Let t ,t ,...,t and n ,n ,...,n be positive integers, Σ = s (t −1) and
1 2 s 1 2 c i=1 i
P
r = R(S ,S ,...,S ). If n ≥ n ≥ ··· ≥ n , then
t1 t2 ts 1 2 c
c
R(S ,S ,...,S ,n K ,n K ,...,n K ) = max{r +δ,n }+ (n −1)+1,
t1 t2 ts 1 2 2 2 c 2 1 X i
i=1
where δ = 0 ifΣ < max{n ,2n },Σ iseven andsomet is even,and δ = −1, otherwise.
1 2 i
2
As an easy corollary of Theorem 1.2, we have the following result which generalizes a
knownresultofGya´rfa´s and Sa´rko¨zy [6]on theRamsey numberofonestarversustwo stripes.
Corollary1.3. Let t ≥ 1 andn ≥ n ≥ ··· ≥ n bepositiveintegers. Then
1 2 c
c
R(S ,n K ,n K ,...,n K ) = max{t,n }+ (n −1)+1.
t 1 2 2 2 c 2 1 X i
i=1
By Corollary 1.3, fort ≤ n , n = max{n ,n ,...,n },wehave
1 1 1 2 c
c
R(S ,n K ,n K ,...,n K ) = R(n K ,n K ,...,n K ) = n + (n −1)+1,
t 1 2 2 2 c 2 1 2 2 2 c 2 1 X i
i=1
which strengthens significantly a well-known result of Cockayne and Lorimer on the Ramsey
number of stripes. In the other word, if G is a graph obtained by deleting the edges of a graph
with maximumdegree (n −1) from a completegraph on R(n K ,n K ,...,n K ) vertices,
1 1 2 2 2 c 2
then
G → (n K ,n K ,...,n K ).
1 2 2 2 c 2
In addition, we obtain the following interesting result if we investigate to Corollary 1.3,
when t ≥ n .
1
Corollary1.4. Let n ≥ n ≥ ··· ≥ n bearbitrarypositiveintegers, andlet G be a graphon
1 2 c
n ≥ n + c (n −1)+1 vertices suchthatδ(G) ≥ c (n −1)+1. Then
1 i=1 i i=1 i
P P
G → (n K ,n K ,...,n K ).
1 2 2 2 c 2
Proof. Set t = n− c (n −1)−1. Clearly t ≥ n andso byCorollary 1.3,we have
i=1 i 1
P
R(S ,n K ,n K ,...,n K ) = n.
t 1 2 2 2 c 2
Since G has n vertices and δ(G) ≥ c (n −1) +1, we have ∆(G¯) ≤ t−1, which means
i=1 i
thatG¯ is aS -free graph and sotheaPssertionholdsbytheaboveequation. (cid:4)
t
It is also worth noting that the condition on the minimum degree in Corollary 1.4 is best
possible. Indeed, let G be a graph on n ≥ R(n K ,n K ,...,n K ) vertices whose vertex
1 2 2 2 c 2
set is partitioned into disjoint sets A, B with |A| = Λ = c (n − 1), |B| = n− Λ and let
i=1 i
P
E(G) = {uv|{u,v}∩A 6= ∅}. Now, set V = B and consider a partition of vertices of A into
0
sets V ,V ,...,V of sizes n − 1, n − 1, ...,n − 1, respectively. For each i = 1,2,...,c
1 2 c 1 2 c
colorwiththei-th coloralledges withinV oredges withonevertexinV and onein V , where
i i j
j < i. In this coloring, the largest monochromaticmatching of color i has n −1 edges, while
i
theminimumdegreeofG isΛ.
3
2 Proof of Theorem 1.2
InordertoproveTheorem1.2,weneedsomelemmas. First,westartwiththefollowingsimple
butuseful lemma.
Lemma2.1. Lett ,t ,...,t bepositiveintegers,Σ = s (t −1)andletH beagraphwith
1 2 s i=1 i
P
χ′(H) ≤ Σ. ThenE(H)canbedecomposedintoedge-disjointsubgraphsH ,H ,...,H such
1 2 s
that∆(H ) ≤ t −1.
i i
Proof. Consideraproperedge-coloringofH withχ′(H)colors. Partitionthesetofcolorsinto
ssetsA ,A ,...,A ofsizesatmostt −1,t −1,...,t −1,respectively. LetH ,1 ≤ i ≤ s,be
1 2 s 1 2 s i
thesubgraphofH inducedbytheedges ofcolorsinA . Clearly H ’sarethedesiredsubgraphs
i i
whichdecomposeE(H). (cid:4)
An alternating cycle in an edge colored graph is a cycle which is properly colored i.e. no
two consecutive edges in the cycle have the same color. We say that a vertex v in an edge
coloredgraph Gseparatescolorsifno componentofG−v isjoinedtov byat leasttwoedges
of different colors. Grossman and Ha¨ggkvist gave a sufficient condition under which a two-
edgecolored graphmusthaveanalternatingcycle. In [5]Grossmanand Ha¨ggkvistprovedthat
if G is a graph whose edges are colored red and blue and there is no alternating cycle in G,
then G contains a vertex v that separates the colors. Bang-Jensen and G. Gutin asked whether
Grossman and Ha¨ggkvist’s result could be extended to edge-colored graphs in general, where
there is no constraint on the number of colors. In [9] Yeo gave an affirmative answer to this
questionasfollows.
Theorem 2.2. ([9]) If G is a c-edge-colored graph, c ≥ 2, with no alternating cycle, then there
is a vertex v ∈ V(G) such that no connected component of G−v is joined to v with edges of
morethanonecolor,i.eG containsavertexseparatingcolors.
Let t ,t ,...,t be positiveintegers,Σ = s (t −1) and r = R(S ,S ,...,S ). Also
1 2 s i=1 i t1 t2 ts
let n ≥ n ≥ ··· ≥ n bepositiveintegersanPdΛ = c (n −1). Set
1 2 c i=1 i
P
max{r,n }+Λ+1 Σ and atleast onet is even and
1 i

Σ < max{n ,2n },
f(t ,t ,...,t ,n ,n ,...,n ) =  1 2
1 2 s 1 2 c 
max{r −1,n }+Λ+1 otherwise.
 1

Infact,f(t ,t ,...,t ,n ,n ,...,n )isthenumberthatweclaimedisequaltotheRamsey
1 2 s 1 2 c
numberR(S ,S ,...,S ,n K ,n K ,...,n K )inTheorem1.2. Usingthesenotations,we
t1 t1 ts 1 2 2 2 c 2
havethefollowinglemma.
Lemma 2.3. Let t ,t ,...,t and n ,n ,...,n with n ≥ n ≥ ··· ≥ n be positiveintegers
1 2 s 1 2 c 1 2 c
andletGbeagraphonf(t ,t ,...,t ,n ,n ,...,n )verticessuchthatG¯ 9 (S ,S ,...,S ).
1 2 s 1 2 c t1 t1 ts
Then
G → (n K ,n K ,...,n K ).
1 2 2 2 c 2
4
Proof. Assume that the statement of this lemma is not correct and suppose that a counterex-
ample exists. Therefore, there are some positive integers t ,t ,...,t and n ,n ,...,n with
1 2 s 1 2 c
n = max{n ,n ,...,n }, and agraph G onf(t ,t ,...,t ,n ,n ,...,n ) vertices,such that
1 1 2 c 1 2 s 1 2 c
G¯ 9 (S ,S ,...,S ) and G 9 (n K ,n K ,...,n K ).Notethat c ≥ 2, byTheorem 1.1.
t1 t1 ts 1 2 2 2 c 2
Among all counterexamples let G be a minimal one having the maximum possiblenumber
ofedges, i.e. G isa graphsatisfies thefollowingconditions:
(a)Thenumberofvertices ofG,f(t ,t ,...,t ,n ,n ,...,n ), isas smallaspossible.
1 2 s 1 2 c
(b) Among all counterexamples satisfying (a), G is a counterexample with minimum c, i.e. no
counterexampleiscolored withlessthan ccolors.
(c) Among all counterexamples satisfying (a) and (b), G is one having the maximum possible
numberofedges.
The fact G 9 (n K ,n K ,...,n K ) implies that the edges of G can be colored by colors
1 2 2 2 c 2
β ,β ,...,β so that for each i, 1 ≤ i ≤ c, the induced graph on edges of color β does not
1 2 c i
contain a subgraph isomorphic to n K . Let Gi be the subgraph of G induced by the edges of
i 2
color β . As |V(G)| ≥ R(n k ,n K ,...,n K ), we deduce that G is not a complete graph.
i 1 2 2 2 c 2
Let u,v be non-adjacent vertices in G. As G satisfies (a), (b) and (c), n K ⊆ Gi +uv (to see
i 2
this,itonlysuffices toadd theedgeuv toG and coloruv byβ and thenusetheproperty (c)of
i
G) which means that (n − 1)K ⊆ Gi. Let M be the matching of size (n −1) in Gi. Since
i 2 i
n K * Gi, we must have Ni(u),Ni(v) ⊆ V(M). Moreover, the fact n K * Gi implies that
i 2 i 2
for each edge xy ∈ M, the number of edges of color i between {x,y} and {u,v} is at most 2.
Thusdegi(u)+degi(v) ≤ 2(n −1),foreach i = 1,2,...,c. Therefore,
i
c
deg (u)+deg (v) = (degi(u)+degi(v)) ≤ 2Λ. (1)
G G X
i=1
SinceG¯ 9 (S ,S ,...,S ),thereisascoloringofedgesofG¯ suchthatthegraphinduced
t1 t1 ts
by thei-th color does notcontain S as asubgraph. Thus,for every vertexv ∈ V(G), wehave
ti
deg (v) ≤ Σ. (Indeed, if v is a vertex with deg (v) ≥ Σ + 1, then the Pigeonhole principle
G¯ G¯
implies that any s coloring of the edges of G¯ contains a monochromatic S of i-th color with
ti
centerv,forsomei,acontradiction). Therefore,δ(G) ≥ f(t ,t ,...,t ,n ,n ,...,n )−Σ−1.
1 2 s 1 2 c
An easy calculation shows that δ(G) ≥ Λ+1 unless Σ ≥ max{n ,2n }, Σ is even and some
1 2
t is even and in this case, we have δ(G) ≥ Λ. If δ(G) ≥ Λ+1, then for every pair of vertices
i
u,v, deg (u) + deg (v) ≥ 2(Λ + 1). Using (1), we deduce that a counterexample could not
G G
exist unless Σ ≥ max{n ,2n }, Σ is even and some t is even. Therefore, hereafter we may
1 2 i
suppose that Σ and at least one t is even and Σ ≥ max{n ,2n }. Note that, in this case we
i 1 2
have f(t ,t ,...,t ,n ,n ,...,n ) = Σ+Λ+1. By (1) and the fact δ(G) ≥ Λ we conclude
1 2 s 1 2 c
thatforevery pairofnon-adjacent verticesu,v inG:
deg(u) = deg(v) = Λ, (2)
degi(u)+degi(v) = 2(n −1), i = 1,2,...,c (3)
i
5
Claim1. Σ ≤ Λ.
Proofof Claim1. On thecontrary,let Σ ≥ Λ+1. Itis easy toseethat
R(S ,S ,...,S ,n K ,n K ,...,n K ) ≤ R(S ,S ,...,S ,(Λ+1)K ).
t1 t2 ts 1 2 2 2 c 2 t1 t2 ts 2
As Σand somet are even,byTheorem 1.1wehave
i
R(S ,S ,...,S ,(Λ+1)K ) = Σ+Λ+1,
t1 t2 ts 2
whichimpliesthat
R(S ,S ,...,S ,n K ,n K ,...,n K ) ≤ Σ+Λ+1.
t1 t2 ts 1 2 2 2 c 2
This means that K → (S ,S ,...,S ,n K ,n K ,...,n K ), where N = Σ + Λ + 1.
N t1 t2 ts 1 2 2 2 c 2
Therefore G¯ → (S ,S ,...,S ) orG → (n K ,n K ,...,n K ), acontradiction. (cid:3)
t1 t1 ts 1 2 2 2 c 2
Claim2. Gis a2-connected graph.
Proof of Claim 2. By Claim 1, one can easily check that δ(G) ≥ Λ ≥ |V(G)|, unless Σ = Λ.
2
Therefore, by the Dirac’s Theorem [1], G is a hamiltonian graph and so a 2-connected graph
unlessΣ = Λ. Now,assumethatΣ = Λ. Inthiscase,|V(G)| = 2Λ+1andδ(G) ≥ Λ. Clearly,
G is connected (in fact the diameter of G is two, since every two non-adjacent vertices have
a common neighbor). If there is a cut vertex v of G, then G − v has exactly two components
G ,G with
1 2
G[V(G )∪{v}] = G[V(G )∪{v}] = K .
1 2 Λ+1
Now,we claimthat all edges ofG ∪{v}(also G ∪{v})havethesamecolor. To seethis,
1 2
let v be an arbitrary vertex of G and let the edge vv is colored by β , for some i, 1 ≤ i ≤ c.
1 1 1 i
Since G ∪ {v} is a complete graph and v is an arbitrary vertex of G , in order to show that
1 1 1
all edges of G ∪{v} have the same color, it only suffices to show that all edges of G ∪ {v}
1 1
incident to v are of color β . On the contrary, assume that the edge v v of G ∪ {v} is of
1 i 1 2 1
color β , where j 6= i. Now let M ,M be arbitrary perfect matchings in G ,G , respectively,
j 1 2 1 2
wherev v ∈ M . Therefore, |M ∪M | = Λandwemayassumethatforeach t = 1,2,...,c,
1 2 1 1 2
the matching M ∪ M contains exactly n − 1 edges of color β , since otherwise for some
1 2 t t
1 ≤ i ≤ c, G has a monochromaticmatching of size n with colorβ , which is impossible. Set
i i
M = (M \{v v })∪M ∪{vv }. Clearly M contains a monochromaticmatching of size n
1 1 2 2 1 i
with color β , which is again impossible. By a similar argument, all edges of G ∪ {v} have
i 2
the same color. Therefore at most two colors are appeared on the edges of G, say β and β
i j
(for some i and j). Without any loss of generality, we may assume that all edges within G
2
are of color β and j 6= 1. As Λ = Σ ≥ max{n ,2n } and |V(G )| = Λ, we obtain that
j 1 2 2
|V(G )| ≥ 2n ≥ 2n which means that G contains a subgraph isomorphic to n K of color
2 2 j 2 j 2
β ,a contradiction. (cid:3)
j
Now the analysis depends on the study ofcertain cycles in G. These are alternating cycles,
coloredwithsomecolorsβ ∈ {β ,β ,...,β },havingnotwoadjacentedgesofthesamecolor.
1 2 c
Therest oftheproofisdevotedtoprovethat an alternatingcycleexistsin G.
6
Claim3. Ghas an alternatingcycle.
ProofofClaim3. Onthecontrary,assumethatGdoesnothaveanalternatingcycle. Thususing
Theorem 2.2, G has a vertex v separating colors. Since G is 2-connected by Claim 2, all edges
ofG incidenttov havethesamecolor,say β . Set G′ = G\{v}. Notethat
i
|G′| = f(t ,t ,...,t ,m ,m ,...,m ) = Σ+Λ′ +1,
1 2 s 1 2 c
where Λ′ = Λ − 1, and m ,m ,...,m are the numbers n ,n ,...,n − 1,...,n in the de-
1 2 c 1 2 i c
creasing order. Clearly any s-coloring of the edges of G¯ induces an s-coloring of the edges
of G¯′. Therefore G¯′ 9 (S ,S ,...,S ). From the minimality of G we deduce that G′ has a
t1 t1 ts
subgraph M isomorphic to (n −1)K whose edges are colored by β . If the degree of v as a
i 2 i
separatorvertexisatleast2n −1,thenthereisavertexu ∈ N (v)whichisunsaturatedbythe
i G
vertices of the matching M. Thus adding the edge uv to the matching M yields a monochro-
matic copy of n K with color β in G, which is impossible. Therefore, the proof of the claim
i 2 i
willbecompletedifweprovethat thedegreeofv as aseparatorvertexisat least 2n −1.
i
First let all edges of G incident to v havecolor β and i ≥ 2. Since Λ ≥ Σ by Claim 1, and
i
alsodeg (v) ≥ δ(G) ≥ Λ,weobtainthatdeg (v) ≥ Σ ≥ max{n ,2n } ≥ 2n ≥ 2n andwe
G G 1 2 2 i
are done.
Now,letalledgesofGincidenttov havecolorβ . Letdeg (v) = 2n −k,forsomek ≥ 2.
1 G 1
Note that the fact Σ ≥ max{n ,2n } implies that |V(G)| = Σ+Λ+1 > 2n and so v is not
1 2 1
adjacent to all vertices of G. Therefore, by (2) we obtain that deg (v) = 2n − k = Λ. This
G 1
meansthat
c
n = (n −1)+k −1. (4)
1 X i
i=2
Since deg (v) = Λ and |V(G)| = Σ+Λ+1, the vertex v has exactly Σ non-neighbors in G.
G
LetS bethesetofnon-neighborsofvinG. By(3),foreveryvertexz ∈ S,deg1(z) = k−2and
fori = 2,3,...,cwehavedegi(z) = 2n −2. Sinceforeveryvertexz ∈ S,deg2(z) = 2n −2,
i 2
Equation (3) implies that the graph induced by the vertices of S is a complete graph. Now, we
prove that G[S] contains an alternating cycle. By Theorem 2.2, G[S] contains an alternating
cycle unless there is a vertex which separates colors. Let z be a vertex of G[S] separating
colors and all edges of G[S] incident to v have the same color, say β . If i = 1 then k − 2 ≥
i
deg1 (z) ≥ Σ−1 which implies that k −1 ≥ Σ ≥ n , which contradicts (4). If i ≥ 2 then
G[S] 1
2n −2 ≥ degi (z) ≥ Σ−1 which impliesthat 2n −1 ≥ Σ ≥ 2n ≥ 2n , a contradiction.
i G[S] i 2 i
Thiscontradictionshowsthat ifv isa vertexofG separating colorsand alledges ofG incident
to v havethesamecolorβ , then thedegreeofv as a separatorvertexis at least2n −1, which
i i
completestheproofoftheClaim3. (cid:3)
Now let C be an alternating cycle of G which has l edges colored by β , for each i =
i i
1,2,...,c, then it has c l vertices. For each i, the l edges of C colored by β form a
i=1 i i i
P
subgraphisomorphicto l K . IfG′ = V(G)\V(C), thenthenumberofverticesin G′ is
i 2
c
f(t ,t ,...,t ,m ,m ,...,m ) = Σ+ (n −l −1)+1,
1 2 s 1 2 c X i i
i=1
7
where m ,m ,...,m are the numbers n − l for i = 1,...,c in the decreasing order. As
1 2 c i i
n ≥ m = max{m ,m ,...,m } and C is a subgraph of G which is properly colored, from
1 1 1 2 c
theminimalityofGwededucethatG′hasamonochromaticsubgraphisomorphicto(n −l )K
i i 2
whose edges are colored by β , for some 1 ≤ i ≤ c. Combining this with a monochromatic
i
subgraph l K of colorβ in C, weobtain a subgraph isomorphicto n K with colorβ in G, a
i 2 i i 2 i
contradiction. Thiscontradictionshowsthatthislemmaistrueand sotheproofiscompleted.
(cid:4)
Now, we are ready to give a proof for Theorem 1.2 which provides the exact value of the
RamseynumberR(S ,S ,...,S ,n K ,n K ,...,n K ).
t1 t2 ts 1 2 2 2 c 2
ProofofTheorem1.2. ToseethattheRamseynumbercannotbelessthantheclaimednumber,
firstconsiderthecasethatΣ < max{n ,2n },Σisevenandsomet iseven. SinceΣandsome
1 2 i
t are even, r = Σ+1 by Theorem 1.1. If Σ < n , then considera partition ofn +Λ vertices
i 1 1
intosetsV ,V ,...,V ofsizes2n −1,n −1,...,n −1respectively. Colorwiththefirstcolor
1 2 c 1 2 c
alledgeswhichareincidentwithtwoverticesofV andforeachi = 2,...,ccolorwiththei-th
1
colorall edgeshavingtwoverticesin V oronevertexin V and oneinV wherej < i. Clearly,
i i j
for each i = 1,2,...,c, the graph induced by the edges of the i-th color does not contain a
subgraphisomorphicto n K .
i 2
If n ≤ Σ < 2n , then partition Σ + Λ + 1 vertices into sets V = A ∪ B, V = C ∪ D,
1 2 1 2
V ,...,V with|A| = |B| = Σ,|C| = 2n −Σ−1, |D| = n −n and|V | = n −1,3 ≤ i ≤ c.
3 c 2 1 2 i i
Color all edges contained in B,D and edges in [B,C],[D,C],[B,D],[A,D] by the first color
β , all edges contained in A,C and edges in [A,C] by β . For each i = 3,4,...,c, color with
1 2
β all edges having two vertices in V or one vertex in V and one in V where j < i. Clearly,
i i i j
foreach i = 1,2,...,c,thegraph inducedbytheedges ofcolorβ doesnotcontainasubgraph
i
isomorphicton K . Theremaininguncolorededgesare[A,B]whichformacopyofK . By
i 2 Σ,Σ
Lemma 2.1, the edges of K can be colored by s-colors α ,α ,...,α such that the induced
Σ,Σ 1 2 s
graph on edges of color α , 1 ≤ i ≤ s, does not contain S as a subgraph. This yields an edge
i ti
coloringofthecompletegraphonmax{r,n }+Λverticeswiths+ccolorsα ,α ,...,α and
1 1 2 s
β ,β ,...,β such that the induced graph on edges of color α , 1 ≤ i ≤ s, does not contain
1 2 c i
S as a subgraph and for each i = 1,2,...,c, the induced graph on edges of color β does not
ti i
contain a subgraph isomorphic to n K . This observation shows that if Σ < max{n ,2n }, Σ
i 2 1 2
isevenand somet iseven,then
i
R(S ,S ,...,S ,n K ,n K ,...,n K ) ≥ max{r,n }+Λ+1.
t1 t2 ts 1 2 2 2 c 2 1
Now assume that the case “Σ < max{n ,2n }, Σ is even and some t is even” does not
1 2 i
occur. Consider a partition of n = max{r −1,n }+Λ vertices into sets V ,V ,V ,...,V of
1 0 1 2 c
sizes max{r − 1,n },n − 1,n − 1,...,n − 1 respectively. For each i = 1,2,...,c, color
1 1 2 c
with β all edges within V or edges with one vertex in V and one in V , where j < i. Now, if
i i i j
r − 1 ≤ n , then |V | = n , and in this case color all edges within V by β . In fact this is a
1 0 1 0 1
c-edge coloring of K that does not have a matching of size n of color β , 1 ≤ i ≤ c. If
n1+Λ i i
r−1 > n ,then|V | = r−1andsothereisanedgecoloringofK withscolorsα ,...,α
1 0 r−1 1 s
without a monochromatic copy of S of color α , 1 ≤ i ≤ s. This yields an (s + c)-edge
ti i
coloring of K that does not have a monochromatic star S with color α , 1 ≤ i ≤ s, and no
n ti i
8
monochromaticmatchingofsizen incolorβ , 1 ≤ i ≤ c. Therefore
i i
R(S ,S ...,S ,n K ,n K ,...,n K ) ≥ f(t ,...,t ,n ,...,n ).
t1 t2 ts 1 2 2 2 c 2 1 s 1 c
Toprovetheotherdirection,consideracompletegraphonN = f(t ,t ,...,t ,n ,n ,...,n )
1 2 s 1 2 c
vertices whose edges are arbitrarily colored by s + c colors α ,α ,...,α and β ,β ,...,β .
1 2 s 1 2 c
Let G bethegraph induced by all edges ofcolorβ ,β ,...,β in K . If foreach i, 1 ≤ i ≤ s,
1 2 c N
the subgraph induced by the edges of color α in K does not contain a copy of S , then
i N ti
G¯ 9 (S ,S ,...,S ) and so Lemma 2.3 implies that G → (n K ,n K ,...,n K ). This
t1 t1 ts 1 2 2 2 c 2
meansthatforsomei,1 ≤ i ≤ c,thesubgraphofK inducedontheedgesofcolorβ contains
N i
asubgraphisomorphicto n K , whichcompletestheproofofthetheorem. (cid:4)
i 2
Acknowledgment
Theresearch ofthefirst andsecond authorsare partiallycarried outintheIPM-Isfahan Branch
and inpart supportedrespectivelyby grantsNo. 94050217and No. 94050057,fromIPM.
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