I nt. J. Math. Math. Sci.
Vol. 6 No. 2 (1985) 279-284
279
ON THE STRUCTURE OF A TRIANGLE-FREE
INFINITE-CHROMATIC GRAPH OF GYARFAS
LARRY EGGAN
Mathematical Reviews
Ann Arbor, MI 48109
Illinois State University
Normal, IL 61761
FRANK HARARY
Churchill College
Cambridge CV30DS, England
University of Michigan
Ann Arbor, MI 48109, USA
(Received June 22, 1981 and in revised form September 13, 1982)
ABSTRACT.
Gyarfas has recently constructed an elegant new example of a triangle-
free infinite graph G with infinite chromatic number.
We analyze its structure by
studying the properties of a nested family of subgraphs G
KEY WORDS AND PHRASES:
Triangle-free, infinite-chromatic
1980 MATHEMATICAL SUBJECT CLASSIFICATION CODE:
i.
whose union is G.
05C15
INTRODUCTION.
Gyarfas [I] described a new example of a triangle-free infinlte-chromatlc graph
G as follows:
1,2
i.e., h
the vertices of G form an
and the vertex v.
of
of
x
matrix, i.e., V
{vi,j,
is adjacent to every vertex of the (i
,v b
{v,Vk,+;
,,k
i, j
+ j)-th column,
,
}.
is easy to see that G is triangle-free for if u,v,w, were vertices of a triangle with
u having the smallest column index, then the fact that uv and uw are edges would mean
.
280
EGGAN and F. HARARY
v and w are adjacent vertices in the same column, which is impossible.
That G
requires infinitely nmny colors follows from Theorem 2 below, although it also
follows directly from the fact that, for j > i, the i-th column contains a vertex
adjacent to all vertices of the
J-th column.
In what follows we will accomplish two things.
We first describe an augmenting
sequence of finite graphs, which has G as its limit, and determine the structure of
This gives a deeper insight into the actual structure of G.
these graphs.
Unless
otherwise specified, we follow the graph theoretic notation and terminology of
Harary [2].
AN ANALYSIS OF G.
2.
In this section we will define a sequence Gn of graphs converging to G.
give some results on the structure of each G
n
G
n
and an alternative way of constructing
which gives a different perspective on its structure.
Finally, we will note that
can be demonstrated by considering subgraphs H
the desired properties of G
We will
(which
are roughly half of G ).
n
For any positive integer n, let Gn be the subgraph of G obtained
DEFINITION.
by removing all vertices v.
x,]
<{vi:j;
subgraph
with i and j greater than n, i.e., G is the induced
n
n}>
< i,J <
of
i G._
First we prove a theorem on the degrees of the vertices of G
n
THEOREM I.
n-
In
k- i
PROOF.
I,
For 0 < k < 2n
2, the number of vertices of G
of degree k is
n
while there are no vertices of degree greater than 2n
Consider G
n
as an n
+ J
n matrix.
Then it is easy to see that
if i < i < n
i,
2.
j
deg(vi,j)
if n
i,
Thus by setting k
k
+
n
+ J
1 or k
j
j
+ i_< i_<
n.
i, we see there are either 2n
i vertices, respectively, of order k, as claimed.
In the next theorem we give the chromatic number x(G
n)_ of each G
n
k
I or
TRIANGLE-FEE INFINITE-CHROMATIC GRAPH
number k
+
Gn
is k-colorable if n < 2
n)
i, that is, x(G
PROOF.
Gn_ I
Since
colorable whereas
G2k
_> I,
For k
THEOREM 2.
G2k
i
+
is a subgraph of
column j so
Ni Nj.
has chromatic
G2k
Gn,
it suffices to show that
is k
G2k_l
To show the latter, suppose on the contrary that
is not.
N. denote the
Now for i < j,
G2k.
while
[log2n].
is colored in k colors, and let
in the j-th column of
k,
281
The sets
Nj
set of colors used on the vertices
is adjacent to every vertex in
vj_i, i
thus form a collection of 2
k
distinct nonempty
subsets of a k-element set, which is impossible.
To show that
G2k_l
1,2,...,2k-i,
j
is k-colorable, let C be a set of k colors and let N.,
be an enumeration of the nonempty subsets of C which is non-
For example, such an enumeration when k
increasing in order of size.
C3
This enumeration provides that if j > i, then there is a color in N
which is not in
N..
3
in
Nr+i;
3 and C
if r
coloring of
+
i > 2
Therefore, color the vertex v
k
i, then use any color in
N.
with a color in
r,l
N..l
i
which is not
This clearly yields a k-
G2k_l.
To conclude this section, we describe an alternative way to construct G
n
we feel gives some insight into its structure and chromatic number.
which
In accordance
with established terminology, we will say that a point covers a set S of points if it
is adjacent to very point of S.
A set T of points smothers S if exactly one point
in T covers S, and T smothers a finite sequence
if there are distinct points t I,
t2,...,t k
S I, S
S
2
k
of sets of points
in T such that t. covers S
for j
1,2
We now describe how to construct Gn using this idea and the join operation +,
where H
+ H’
is the graph obtained from the union of H and
H’;
of H to every point of
stages.
see [2, p.
Here the notation H
H’ + H", and similarly for
+ H’ + H"
21].
H’ by
We describe how to build G
n
Build K
2
in three
stands for the union of two joins H
+ H’
and
more summnds each of which will be a complete graph K
n
or its complement, the totally disconnected graph K
n
STAGE i:
joining every point
+ KI +
Kn_ 2.
(Label the vertex K
r.)
I by
L. EGGAN and F. HARARY
282
STAGE 2:
K by
I
Kn + KI,
I +
K
Sj
Replace the j-th point in
3
Replace the
n
in
and the right
tk3
j
S. by the set of n points T where the adjacency in
3
2
j
T
smothers T
n
n
n
j-kin T J- covers T
for k
1 2
n
n
I,
is preserved, but T
that
aj
bj.)
STAGE 3:
S
numbering from bottom to top, by
(Label the left KI by
n-2.
1,2
for j
Kn_ 2,
T
-i
for j
1,2
n-2.
(Suppose
j-l.)
Figure i shows how the construction progresses when n
when a single isolated point (corresponding to
is
Vn,l
5.
added
This resulting graph,
Gn.
is isomorphic to
We will not formally prove this, though it is easy to see that an isomorphism is
obtained by mapping r to
degree i to
mapping, to
vi, 2,
i
Vk,n+l_j
Vl,l’ bj
n-l,n)
to
Vn_j,l, aj
to
Vn-l-j,2
(and the two vertices of
j
and T to the vertices in column n+l-j with
n
tj ,k
It is also easy to see that another minimal coloring (besides
the one given in the proof of Theorem 2) is obtained by only using colors
Cl’ c2’
ci
to color T
i
j
for i < j < 2 and i
n
n
Stage 1
1,2
k.
5
b3
Stage 3
2
Stage 2
b
b
b
3
2
b
Figure i.
Finally let H be the subgraph of G with the
n
n
{Vl.l’Vl,n} U vi,j
2
j
2
n-l, i
i
1/2(n2-3n+6)
n-J}
vertices
i
TRIANGLE-FREE INFINITE-CHROMATIC GRAPH
It is clear that H
n
is triangle-free and for n
2
k
applying the argument in the
proof of Theorem 2 to columns 2 through n shows that H
though H
is simpler then G
n
n
283
n
is not k colorable.
AI-
while still retaining the cascading appearance
illustrated by Figure i, H is still not critical.
n
We wish to thank the referee who is responsible for an improvement in our
exposition and for the above proof of Theorem 2.
REFERENCES
i.
Gyarfas, A.
Math.
2.
Harary, F.
Still another triangle-free infinite-chromatic graph, Discrete
30 (1980), 185.
Graph Theory, Addison-Wesley, Reading, 1969.