University of Ljubljana Institute of Mathematics, Physics and Mechanics Department of Mathematics Jadranska 19, 1000 Ljubljana, Slovenia

Preprint series, Vol. 36 (1998), 601

THE GRAY GRAPH REVISITED

Dragan Maruˇsiˇc TomaˇzPisanski

ISSN 1318-4865

May 8, 1998

Ljubljana, May 8, 1998

THE GRAY GRAPH REVISITED



Dragan Marusic Tomaz Pisanski

IMFM Oddelek za matematiko IMFM Oddelek za matematiko

Univerza v Ljubljani Univerza v Ljubljani

Jadranska Ljubljana Jadranska Ljubljana

Slovenija Slovenija

draganmarusicuniljsi tomazpisanskifmfunilj si

Abstract

Certain graphtheoretic prop erties and alternative denitions of

the Gray graph the smallest known cubic edge but not transitive

graph are discussed in detail

Intro duction history

The smallest known cubic edge but not vertextransitive graph has

vertices and is known as the Gray graph denoted hereafter by G The rst

published account on the Gray graph is due to Bouwer who mentioned

that this graph had in fact b een discovered by Marion C Gray in thus

explaining its name Bouwer gives twoways of constructing G First three

copies of the complete K are taken and to a particular edge



e of K a vertex is inserted in the interior of e in each of the three copies



of K and the resulting three vertices are then joined to a new vertex



The second construction identies a particular Hamilton in G and the

corresp onding chords see Figure

Some other w ays of constructing G are presented in this note thus shed

ding a new light on the structure of this remarkable graph In the computa

tions involved a usage of the Vega package was essential

Supp orted in part by Ministrstvo za znanost in tehnologijo Slovenije pro jno P

Supp orted in part by Ministrstvo za znanost in tehnologijo Slovenije pro j no

J and J

Figure The Gray Graph with an identied Hamilton cycle as in

Structural prop erties and alternative de

nitions

The Gray graph G is a cubic bipartite and edge but not vertextransitive

graph Its automorphism group Aut G acts transitively on each of the bipar

tition sets Since its is it is the of two dual trianglefree

p oint line and agtransitive nonselfdual congurations



There is a simple reason for intransitivity of Aut G on the vertex set of

G Two vertices have the same sequence if and only if they b elong

to the same bipartition set More precisely the two distance sequences are

and and the resp ective vertices are

henceforth called black and white It follows that the diameter of G is Note

 

that jAut Gj Let S denote an arbitrary Sylow subgroup



ZZ of Aut G It may b e seen that S o ZZ and that S acts transitively





on the edge set of G as well as on the sets of black and white vertices

We note further that there are a total of o ctagons that is induced

in G Octagons in G play an essential role in our Con cycles of length

struction below Let us also mention that each vertex of G is contained

in o ctagons and each edge of G is contained in o ctagons

Construction This construction was p ointed to us by Randicinaper

sonal communication but see also and gives G in the LCF notation



as the graph with co de thus identyng a Hamilton

cycle which admits a ZZ symmetry see Figure



Figure The Gray graph with an identied Hamilton cycle admitting a ZZ





Figure The Gray graph is a ZZ regular cover of K The broken lines carry identity





voltages whereas Black and white vertices of K lift to the



vertices with distance sequences and resp ectively

Figure The Gray graph is a ZZ regular cover of the Pappus graph P The broken



lines carry identity voltages whereas Black and white vertices of P lift to the

vertices with distance sequences and resp ectively

Construction Following the construction using three copies of K



from the previous section it may be deduced that G is a regular cover of



K with ZZ as the group of covering transformations see for notation





and terminology More precisely letting and be the

 

two generators of ZZ then Figure gives G as a ZZ regular cover of K



 

We note that by averaging the voltages that is by replacing each



and by and by selecting instead of the group h ZZ its i



ZZ as a voltage group the resulting voltage graph K subgroup h i

 

lifts to a fold cover P on vertices which is isomorphic to the Levi graph

of the wellknown Pappus conguration Note that P is a arctransitive



bipartite graph and is the underlying graph of the voltage graph depicted in

Figure Furthermore the voltage graph in Figure lifts to the Graygraph

G Let us also mention that the normalizer in Aut G of the corresp onding



two groups of covering transformations ZZ and ZZ is a subgroup of order





 

Construction Another interesting construction identies G as the anti

line graph of a certain Cayley graph of the Sylow subgroup S of Aut G

Each element of S can be represented as a quadruple i j k l where

i j k l ZZ and the multiplication ob eys the following rules



Figure The line graph LG is a Cayley graph for the Sylow subgroup S



o ZZ Aut G Note that S acts transitively on the edges and the two bipartition ZZ





sets of G

i j k rstw i rj s k t w

i j k rstw i t j rk s w

i j k rstw i s j t k rw



Note that the normal subgroup in S isomorphic to ZZ consists of all the



elements i j k S ijk ZZ Let a and b



Then the Cayley graph CayS fa a bb gof S with resp ect to the

set of generators fa a bb g is the line graph LG of the Gray graph see

Figure

Figure The Gray graph and ve disjoint o ctagons

Construction Finally we want to discuss by far the most interesting

rule for constructing the Gray graph which shows that G p ossesses a surpris

ing feature of containing itself within itself In what follows we shall rst

discuss some graphtheoretic distinction between the two kinds of vertices

and then gradually build our waytowards a construction of G identifying the

Gray graph within the Gray graph feature

Now let x be an arbitrary vertex in G and let G x i i where

r

i i denote the subgraph of G induced by all the vertices at

r

distance i i from x Of course since the vertex orbits of Aut G coincide

r

with the color classes the graph G x i i may only dep end on the color

r

of x

An unordered triple of o ctagons may b e asso ciated with an arbitrary ver

tex in G in the following way It transpires that for a blackvertex b the graph

G b is isomorphic to a union of three o ctagons C these o ctagons give



Figure The Gray graph as a union of G G and G

  

rise to the ab ove triple Similarly for a white vertex w the graph G w

is isomorphic to a union of three disjoint o ctagons and four isolated vertices

C K Again the three o ctagons give rise to the ab ovementioned triple



Further it may be seen that the black and white triples have at most one

o ctagon in common which happ ens if and only if the corresp onding two

vertex set vertices are neighb ors in G It thus follows that the graph whose

consists of all the white and black triples of o ctagons with the adjacency

meaning that two triples have nonempty intersection is isomorphic to the

Gray graph Consequently the graph whose vertex set consists of all oc

tagons in G with two o ctagons adjacent if and only if they b oth belong to

one of the ab ove triples is isomorphic to the line graph LG of the Gray

graph Figure identies ve disjoint o ctagons the upp er three o ctagons

dene a black triple whereas the top o ctagon together with the two b ottom

o ctagons dene a white triple

discussion the rule for constructing G may be given Based on the ab ove

via three auxiliary graphs G G b G G b G

  

G b shown in Figure The ro ot blackvertex in G is lab eled The



six vertices and at distance from are glued to the

corresp onding vertices of G which is depicted in the pro jective plane



The twelve vertices a a a a b b b b c c c c at distance from

        

are identied with the corresp onding twelve midp oints of the sub divided

cub e G Three disjoint grey o ctagons E E E are visible on the pro jective

 a b c

plane The faces of the sub divided cub e give rise to three opp osite disjoint

o ctagonal pairs A A B B C C

Figure The lo cal situation in the o ctagon graph isomorphic to LG

Figure depicts the lo cal situation in the ab ove mentioned graph of

o ctagons isomorphic to LG where as seen from the graph G in Figure



the adjacency corresp onds to two o ctagons being at distance in G For

instance the o ctagon E is at distance from E and E as well as from

a b c

octagons C and C see also the ve shaded o ctagons in Figure

References

I Z Bouwer An edge but not vertex transitive Bul l Can Math Soc

R Frucht A canonical representation of trivalent Hamiltonian graphs J Graph

Theory

JLGross and TW Tucker Top ological AddisonWesley

T Pisanski M Randic Bridges b etween geometry and graph theory to appear

T Pisanski ed Vega Version Quick Reference Manual and Vega Graph

Gallery Ljubljana httpvegaijpsi