AbstractWe prove that there are exactly two connected graphs which are locally a cube: a graph on 15 vertices which is the complement of the (3×5)-grid and a graph on 24 vertices which is the 1-skeleton of a certain 4-dimensional regular polytope called the 24-cell

Buset, Dominique

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Graphs which are locally a cube 

We prove that there are exactly two connected graphs which are locally a cube: a graph on 15 vertices which is the complement of the (3×5)-grid and a graph on 24 vertices which is the 1-skeleton of a certain 4-dimensional regular polytope called the 24-cell. © 1983.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

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Graphs which are locally a cube

Discrete Mathematics 46 (1983) 221-226 
North-Holland 
@RAPES VWHN!H ARE LOCALLY A CUBE 
Dominique BUSET 
Received 29 October 1982 
We prove that there are exactly two connected graphs which are locally a cube: a graph on :I5 
vertices which is the complement of the (3 x 5)-grid and a graph on 24 vertices which is the 
l-skeleton of a certain 4-dimensional regular polytope called the 24-cell. 
Let (e,, e2, e3, e4) be the standard basis of R4. The 24-ceI& is a 4-dimensional 
regular polytope whose vertices are the 24 vectors *q =te, (i # i) of R4, two 
vertices being adjacent iff the angle between the corresponding vectors is 60’. It is 
well known [2] and easy to check that the l-skeleton (i.e. the graph consisting of 
the vertices and edges) of this polytope has the following property: for every 
vertex u, the neighbourhood of u (i.e. the subgraph induc& by G on the set of 
vertices adjacent to V) is isomorphic to the l-skeleton of a 3-dimensional cube. In 
other words, the 24-cell1 is locally a cube. 
More generally 133, given a graph G’, we shall say that a graph G is locally G’ 
if, for evey vertex t, of G, the neighbourhood G(u) of u is isomorphic to 6’. If G’ 
is the l-skeleton of a 3-dimensional cube and if G is locally G’, we shall say that 
G is locally a cube. 
It is natural to ask whether the 24-cell is the only connected graph which is 
locally a cube. 
‘Ikom. If a connected graph G is locally a cube, G is isomorphic either to the 
l-skeleton of the 24-cell or to the complement of the (3 x 5)-grid. 
The (p X&grid is the graph whose vertices are the pq ordered pairs (i, j) witEh 
i=l ,..., p and i=l,..., q, two vertices being adjacent iff they have onie 
coordinate in common. 
The adjacency relation in a graph G will be denotea by - and the number of 
vertices of G by IGI. 
2. Proof of the Iheorem 
Lemma 1. For any two adjacent vertices of G, there are exactly 3 vertices of G 
adjacent to both of them. 
0012-365x/83/$3.00 @ 1983, Elsevier Science Pub’lishers B.V. (North-Holland) 
222 D. Bwe1 
Prost. This follows immediately from the fact that the cube is a regular graph of 
degree 3. 
Let u be a fixed vertex of G. Mk shall denote by Vi (i = 1,. . . ,8) the vertices of 
G(u) and by Gi the su’bgraph induced by G on the set of vertices adjacent to q 
and at distance 2 from o. Since the neighbourhood of Vi is isomorphic to a cube, 
Gi is a 3-claw, that is Gi has 4 vertices Wi, il, i2, i3 such that wi - & for every 
r=l,2,3 and i& for every tfs. 
LRrrmna 2. If Vi # Uj, thien Gi # Gi and the subgraph Gi fI Gi is not an edge. 
PtoOf. If Gi = Gj, the vertices Oi, I+, il, i2, 3 i are all in the neighbourhood of Wig 
which is isomorphic to a cube. This is a contradiction because tri and Uj are both 
adjacent to il, i2, i3 and the graph of a cube cannot contain 5 such vertices. 
If the claws Gi and Gj have exactly one edge in common, we may assume 
without loss of generality that it is the edge (Wi. iI}. SO that Wj = Wi or Wj = il. In 
any cae, the neighbourhood G(wi) contains the vertices Ui, Uj, i,, i2, i3 with Uj - il. 
Since G(w,) is isomorphic to a cube, uj must also be adjacent to one of the 
vertices i2 or i3, and SO Gi and Gj have at least tvlo edges in common, 
contradicting the initial assumption. 
Lemmra 3. If ui -I uj, than wi # wj, wi $ Gj* Wj $ Gi and \Gi n Gj I= 2. 
Proob, Since ui -- ui, there are exactly 3 vertices adjacent to Ui and Uj by Lemma 1. 
One of them is u. There is no vertex adjacent to Vi and Uj in G(u). Therefore the 
two missing vertices are at distance 2 from u, and so \Gi n Gil = 2. 
If W, E Gj, then Wi - ui. This contradicts Lemma 1 because Ui and Wi are both 
adjacent to i ,, i2, i3, tlim Therefore Wi $ Gj and similarly Wig Gi. In particular, 
w, # w,. 
Let d(ui, Vi) denote the distance between Vi and Ui in the subgraph G(U). 
Lemma 4. If d( vi, 4) = 2, then Gi n Gj # $3, { Wi} and {Wi). 
kc. If Gi nGi is equal to $9, {Wi} or {wi}. then the vertices i,, i2, i3, jt, j2, j3 are 
painvise distinct. Since d(Ui, Vi) = 2, there is Li vertex uk E G(u) adjacent to Ui and 
u,- By Lemma 3. wk $ Gi U Gi and we may assume without loss of generality that 
G, (7 Gk = {i,, L} and Gj n Gk = {i,, &}. It follows that G(Q) contains at least 9 
verticczs. a contradiction. 
~IIXBUB 5. Zf n( v,, t‘, ) = 2 md wi = w,, then the subgraph Gi n G, is a Z-claw (i.e. 
the union of two intersecting edges). 
kf. This is a direct consequence of Lemmas 2 and 4. 
Graphs whicli me badly a crtbe 223 
Roof. By Lemmas 2 and 4, we already know that IG, n Gil = 1, ‘2 or 3. In view of 
Lemma 3, we may assume that Gi 17 Gk = {it, iz) and Gj Tr Gk = (i1, iz). 
By hypothesis, Wi # wi. Moreover, Wj # il, i2 because wig Gk thanks to Lemma 
3. If Wj = is, then C# iJ for every r, s E (1,2,3} and so il, iz, jl, j2 are 4 distinct 
vertices adjacent to v k. It follows that G(Q) contains at least 9 vertices, a 
contradiction. Therefore ~54 Gi and similarly Wi 6 Gj, so that Gi f’T Gi s (it, i2, is). 
Observe also that (il, iz) $ Gi f? Gj, because otherwise il and i2 would be adjacent 
to Vi, Vi, W& which is a contradiction since il, iz, Vi, vi, wk are all in G(vk) and since 
the graph of a cube cannot contain 5 such vertices. 
(i) If IGi n Gj/ = 3, then Gi f? Gj -~(i~, i2), which is impossible as we have just 
seen before. 
(ii) If (Gi fT Gil = 2, it is no loss of generality, thanks to the preceding observa- 
tions, to assume that Gi n Gj = (il, i3} and Gc = {wk, il, iz, i2) with il = iI. The 
neighbourhood G(i,) COl;ltahX Vi, vi, vk, Wi, Wj, Wk with Wi - Vi - Vk - Vi $ Wi. Since 
G(i,) is isomorphic to a cube, there must be a vertex x E G(i,) adjacent to y and 
vi but not to Wi. Moreover x E G(V) because x - Vi, x # V, x # Wi and x ?_ Wi. Thus 
x = v1 and so vI - il. Using the fact that G(S) is a cube, we get q - i3. Mow, in 
G(q), vi and Vi are both adjacent to V, I ‘1, i3, a contradiction since G(q) is a cL<tie. 
(iii) Therefore IGi n GJ = 1 and Gi n Gj = (4) for some r E (1,2,3}. Together 
with Gi nGk ={il, iz) and Gj n Gk =&, j2), this implies rf 3 and SO, without iOSS 
of generality, Gi n Gj = (iI) = fil) and Gk = ( Wk, il, i2, j2). Using the same type of 
arguments as in (ii), we get v1 - il, and, because G(i,) is a cube, vff Wk. Thus 
w1 # Wk and the vertices Wi, wi, Wk, WI are pairwise distinct. 
Lemma 7. If d(vi, vj) = 3, #ten Wi f Wj. 
Roof. Assume that Wi = Wj and let Vi - v,,, - v,, - vi be a path of length 3 joining 
Vi to Vi in G(V). 
If IGifTGj)=I, then GinGj=(Wi} and i,#i, for every r,s~{1,2,3}. By 
Lemma 3, we may assume that Gi fT G,,, = (iI, i2). Moreover, since w,,, # w, = Wj, 
we may assume, by Lemma 6, that Gj = G,,, = &}. Therefore G,,, = {w,, i,, i2, j3}, 
which implies j3 j- il and j3 # iz, a contradiction in the cube G(Wi). 
If I Gi f7 Gil = 2, then Gi n Gi is an edge, contradicting Lemma 12. 
If IGi n Gj I = 3, then the subgraph Gi f”I Gj is a 2-claw and we may assume that 
Gj z {Wi, il, i2, j3) with il =jl and i2=j2. By Lemma 3, lGifIG,,,I=2 tith n\$G,,,, 
and SO G,,, contains at least one of the two vertices il, iz_ Since w, # wi = wj, 
Lemma 6 implies that Gj fT G,,,, = (is) for some s E (1,2,3}, and so G,,, contains at 
most cae rrf the two vertices il, i2. Therefore, without any loss of generality, 
224 D. Buset 
Gi nG,,, =(i2). Now, by Lemma 6 again, G(i2) has exactly 4 vertices in common 
with the cube G(u), namely oil u, and the two vertices of G(u) adjacent to ui and 
Q,, On the other hand, Ui E G(i2) n G(u). This is a contradiction since uc is not 
adjacent to uj. 
If IGi n Gj I= 4, then Gi = Gi, contradicting Lemma 2. 
FrqosSon 1. If a graph G is Zocally a cube and if, fer some uertex u of G, them 
are two WT&TS ui, ui E G(U) SU& that cl(~)i, ui) = 2 and Wi # W+ then G is isomorphic 
to the l-skeleton of the 24-cell. 
Proof. It is easy to check that the l-skeleton of the 24-cell satisfies the above 
hypothesis. Therefore, it suffices to prove that a graph G satisfying this hypothesis 
is uniquely determined up to &morphism. 
We shah denote the adjacencies in the cube G(u) by 
u1~-u2-t)3-u4-u~, US -tJ6-2)7-U&$-2)5 
and 
ui - vi+4 for every i = 1, :E, 3,4. 
Suppose that wl # We. Then, by Lemma 6, G, n G3 = (11) without any loss of 
generality and ul, u2, u 3r u4. wl, 159~~ w3,w4 are the 8 vertices of the cube C(i,), 
with 
wl-w2-w3--w4-w,, w2.f w4 
Since us- ul and us + 11, it follows from Lemma 3 that G1 n G5 = {12, 13). 
Moreover, by Lemma 3 again, G1 n G2 = { 1 rr 12) without any loss of generality, 
and so G1 n G, = { 1 Ir 1,) thanks to Lemma 6. The neighbourhood G(i,) contains 
the vertices vI, u2, us, wlr w2, w5 with 
and w5 # wl by Lemma 3. Since G(i,) is a cube, we have w2 # w5 and so, by 
Lemma 6, G,n Gs = {12} and ul, u2, us, I+,, wl, w2, ws, )96 are the 8 vertices of 
the cube G(l,), with 
By sunilar arguments, w4 # ws, Gj n Gs = { 13b and ul, u4, us, 2)8, wl, w4, ws, wg 
are the 8 vertices of the cubi: G(13), with 
I 
Wl--w4 - ws.- ws - WI, WI 3- W8, w4 + ws 
The neighbourhood G&) contains the vertices ul, 11, 12., 13, wXr w4, ws. Since 
G(wJ is a cube, the missing vertex w E G(w,) must be adjacent to w2, w4, ws and 
non adjacent to 11, 12, 13, SO that W# WI, W2, W3, W4, w5c W(,, W8. Note that 
W2 + w8 (beCaUSe w2 is already adjacent t0 8 Vertices distinct from wg) and dS0 
w4 L W6, w5 t w3* 
Using similar arguments, it is now easy (but a little bit tedious) to show that the 
subgraph induced by G on the set of vertices wi (i = 1,. . . ,8) is isomorphic to a 
cube which is precisely the neighbourhood G(w). Moreover, given any 4 vertices 
q, 4, N, y in a face of the cube G(o), there is exactly one vertex fi of G which is 
adjacent to Q, vj, t]k, q and to the vertices wi, wj, wk, wt of the corresponding face 
of the cube G(w); for example, we have seen that 11, 12, l3 are three of the 
vertices fi, f-, f3, f4, fS, fa. This shows that the graph G has 1 + 8 + 6+ 8 + 1 = 24 
vertices and, being uniquely determined up to isomorphism by the preceding 
construction, it is isomorphic to the l-skeleton of the 240cell. 
propositIon 2 If (1 gmph G is 20&y Q CUB and if, for some uertex u of G, wi = wj 
whenever 4, uj E G(U) with d(ui, uj) = 2, then G is jsomo@ic to the complement of 
the (3 X 5).grid. 
Proof. It is easy to check that the complement of the (3x5)-grid satisfies the 
above hypothesis. Therefore, it suffices to show that a graph G satisfying this 
hypothesis i  uniquely determined up to isomorphism. 
We use the same notations as in the proof of Proposition 1 to denote the 
adjacencies in G(u). 
The hypothesis implies that w1 is adjacent to up, u3, u6, 218 and that w2 is 
adjacent to u2, u4, us, 2j7, with w1 # w2 by Lemma 3. Using Lemma 5, we may 
assume without loss of generality that G1 ={wl, 11, 12, 13} and G3 = 
{WI, 12, 13, 3,). Since G(w,) is a cube, it follows that 31 is adjacent to u6 and z)$ 
and, without loss of generality, G6= {wl, 11, 13, 3,) and Gs=(wr, l,, 12, 3,). By 
Lemmas 3 and 5, the subgraphs G2, G4, G5, G7 are then completely determined. 
This construction shows that the graph G has 15 vertices and is uniquely 
determined up to isomorphism. 
The proof of the Theorem follows immediately from Propositions 1 and 2. 
3. FinaI comments 
A. Brouwer [l] proved independently that there are exactly two connected 
graphs which are locally a cube. After some exchange of information, he could 
prove a more general result characterizing the graphs which are locally the 
complement of a (p x q)-grid with p aq 22 (q > 2 or p > 3). We shall say that 
these graphs are locally pq. 
‘I%eorem (Browwr [In. If G is a conmxted graph which is locally p x q with 
p 3 q > 2 (q > 2 or p > 3), then G is the complement of a ((p + 1) X (4 + N-grid or 
(i) p = 4, q - ‘2 and G is the l-skeleton of the 24-cell 
(ii) p = q = 3 cd G is the Johnson scheme (2) on 20 vertices (that is the graph 
226 D. &set 
whose w&es are the 3-subsets of (3 6-set, two uedces being adjacent ifi the 
comsponding 3-subsets intersect in a 2-subset). 
The remaining cases (p, CJ) = (3,2), (2,2) or (p, 1) with p > 1 allow infinitely 
many nonisomorphic solutions. K2 is obviously the unique locally m graph. 
[l] A. Brouwer, Personal communication. 
[2] H.S.M. Coxeter, Regular polytopes (Dover Publications, New York, 1973). 
[3) J.I. Hall, Locally Petersen graphs, J. Graph Theory 4 (1980) 173-187. 


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Graphs which are locally a cube

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