AbstractA cubic lattice graph with characteristic n is a graph whose points can be identified with the ordered triplets on n symbols and two points are adjacent whenever the corresponding triplets have two coordinates in common. An L2 graph is a graph whose points can be identified with the ordered pairs on n symbols such that two points are adjacent if and only if the corresponding pairs have a common coordinate. The main result of this paper is two new characterizations and shows the relation between cubic lattice and L2 graphs. The main result also suggests a conjecture concerning the characterization of interchange graphs of complete m-partite graphs

Cook, Curtis R.

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8 DISCRETE MATHEMATICS 4 (1973) 129-I 38. North-Holland Publishing Company 
FURTHER CHARACTERIZATIONS OF CUBIC LATTICE GRAPHS~ 
Curtis R. COOK 
h4WImCN of Mathematics. Oregon State University, Cowallis, Oregon 97331, USA 
Received 15 July 1971 
Ma&act. A cubic kttice graph with characteristic n is a graph whose points can be identified 
with the ordered tripkts on n symbols and two points are adjacent whenever the corresponding 
tripkts have two coordinates in common. An Ls graph is a graph whose points can be identi- 
fied with the ordered pairs on n symbols such that two points are adjacent if and only if the 
correapondi~ pairs have a common coordinate. The main result of this paper is two new char- 
acterizations of cubic kttice graphs. The main result depends on a new Ls graph characteriza 
tion and shows the relation between cubii lattice and Ls graphs. The main result also sqzgests 
a conjecture convening the characterization of interchange graphs of complete m-partite 
Btaphr 
I. I4moduclion 
A cubic lattice graph with characteristic II is a graph whuie points 
can be identified with the ordered triplets on n symbols and two points 
a~ adjacent whenever the corresponding triplets have two coordiiiates 
in Lzmmon. An &, Braph is a graph whose points can be identified with 
the ordered pairs on 14 symbols such that two points are adjacent if and 
only if the corresponding pairs have a common coordinate. The main 
result o!’ this paper is two new characterizations of cubic lattice graphs. 
I’he main result depends on a new L, graph characterization and shows 
the retatisn between cubic lattice and L2 graphs. The main result sug- 
@zsta  conjecture concerning the characterization of interchange graphs 
of complete mqwtite graphs. 
In this paper, we consider only finite undirected graphs without loops 
w multtpk Ii= or edge& The &WC of a point is the number of lines 
in&dent with that point. A gaph is h#uktr if all its points have the same 
m. A p~pb Rcwn~rrd if every pair of points are joined by a path. 
l’k &~~llpr d(rc. u) between points Y and u is the length of a shortest 
130 C.R. Cook. Further chantctetizationr of adtic ~tti@gm#s 
path joining them. The number of points w adjac snt to both 11 and u 
is denoted A(u, u). If II is adjacent to u. A(rc. u) is c lled the edge degree 
of the edge uv. 
Shrikhaode [ 81 and Moon [ 71 showed tha: the following properties 
characterize an L, graph G except for one exceptional case when ?1=4: 
(A, ) G Ins v* pciqts. 
(A? ) G is regular of degree 2(n- I ). 
(A3) If GI(u. u) = I, then A(u. u) = n-2. 
(A4) if d(rr, u) = 2, then A(u. u) = 2. 
Laskar i 5 1 and Aigner I I i showed that tht- following propert& 
characterize a cubic lattice graph G except for one cxccptional case 
when II = 4: 
(B, ) G has rt3 points. 
(B2 ) G is s:onnccPd and regular of degree 3(n- I). 
(B3 ) If (;(d ‘, u) = I , lhen A(tr. c) = II - 2. 
(B,) If d(l&. u) = 2, then A(u. u) = 2. 
(B, 1 IfJ(u. U) = 2, then there exist exactly II- I points IV such that 
cl(rc, IV) = I and J(u. w) = 3. 
Note that (A31 and (BJ). and (A,) and (B4 1 are identical and both 
characterizations have one exceptional case when PI = 4; 
Laskar I61 has also characterized cubic lattihw paphu for I) 3 7 in 
terms of lhe eigenvalues of the adjarcnry matrix. 
2. Qiancteritation of t, mphs 
As mentioned in the introduction. an 6~ gr!?plh is a graph whgsr 
points k.2 I be identified with the HZ ordeti giaitlr on m syn_rWn 31u& 
that twc 1 .Gnts arc adjacent whenever the mrqx&ia pie3 ti a 
common .cxdinate. A mtri.r Ltrrtpli is a paph whose pmtl can be 
identified wrth a set of distinct ordered paift of pcWive intqpa WB 
that two p)ints are adjacent whenever the ~rnzspndittg pstn haw a 
common ctlordinate. Thus an 62 graph i3 a spe&i ca CM a nutfC% 
!j 2. aamctP?hltion of I.2 gaphs 131 
graph. Hedetniemi 141 has obtained four characterizations of matrix 
graphs. The L2 graph characterization follows easily from one of these 
characterizations. This L2 graph characterization will be used in the 
proof of one of the cubic lattice graph characterizations. 
The point set of a graph G will be denoted V(G). A complere graph 
Kp has every pair of its p points adjacent. For any subset S of V(G), 
the induced subgraph (s) has point set S and two points of S are adja- 
cent if and only if they are adjacent in G. Let n be a partition of V(G). 
Then t is a K-partition if the subgraph induced by each block of ?I is 
R complete graph. The lines contained in r are the lines of G contained 
in the subpaphs induced by the blocks of s. Two K-partitions 
*I = t Y, ) . . . . I’,,, 1 and r2 = (WI , .., W,, I are orthogoud if 
I Vi n Wjl < I for every i and i. 
Theorem I (Hcdetniemi). A graph G is a matrix gruph ij’and only if 
there exist two orthclgunal K-partitions of V(G) containing all the lines 
of G. 
Thwrcnr 2. A gtuph C is an L 2 graph if’and only if there exist two 
ortho@rwl Kqvtitians of V(G) containing all the lines of G and each 
pattirim con.Mts of n blocks of order n. 
hf. Let 11 ;f 1 VI, %... V,, i and w2 = (IV,, . . . . IV,, 1 be two orthogonal 
R-partitions of V(G) containing all the lines of G and 1 Vi1 = IlVil = II 
fotl- I , . . ..H. Define a mapping/ from UC) into V(L,) by f(u) = 
tk. nr) if and only if Vt n W,,, - (u 1. Since rI and r2 are orthogonal, 
1 &s I _ I. Ftsrn I v,l t Iw,l t tt for any i and i, it follows that I Vi n Wjl= 
I andfisonto. 
To cxnnplcte the pmf that G L L2 we must shaw that $ preserves 
~~~y,LetctuBcalineinGandIct (10 = Vi” Wiand tu) = 
V, r‘( lu, a Then f(u) s (f, 1) and/(u) - (k. ml). Since the two K-parti- 
tinns rrgntilSn 1 the lines of G. either I = k or j = III. but not both. In 
d&et cjgc (I, D is *tat to (k. nr). 
New aS%me that (I. /I l T(cr) ir adjacent to (S. I) -Mu). This implies 
tkt 4th CI, tt@ bf, s V, w Y, 1) E k> * Wt. Since (Vi) and Cl+‘/) arc com- 
~~~~~~~nt tou. 
I%e mxx&ty SC ths% cxmditions is obvious if WC let the blocks of 
q k the cl)gsnta of k; z with the same Pb coordinate. 
132 C. R. Cook, Further chamcrerizations of cubic lattice paphs 
3. The main theccrem 
Recall that a cubic lattice graph with characteristic n is a graph whose 
pints can be identified with the n3 ordered triplets OII n symbols such 
that two points are adjacent whenever the corresponding triplets have 
two coordinates in common. A cube graph is a graph whose points can 
be identified with a set of distinct ordered triplets of positive integers 
such that two points are adjacent whenever the corresponding triplets 
have two common coordinates. Hence a cubic lattice graph is a special 
case of a cube graph. III [ 21, the author obtained two cube graph char 
acterizations. The two cubic lattice graph characterizations are modi- 
fications of these characterizations. 
A la!tice is a partially ordered set L in which each pair of elements 
has a greatest lower bound and a least upper bound. If LI and /3 are 
two elements in L. we denote their greatest lower bound by a-0 and 
their least upper bound by Q + 0. Let 0 denote the zero element of L. 
The set of partitions of a set S forms a lattice. If L is the set of parti- 
ti.ons of S, then a i b (a -0) if and only if a e b (a) and a I b (B), and 
a p b (a +P) if and only if there exists a sequence ao. a,, . . , at, such that 
a=ao,b=ap,andai=ai+l (a)oraisai+I @)forOsi<p-I. 
Let 1 be a partition of the points of a graph G. Then t is an L, (M) 
partition if the subgraph induced by each block of R is an L, (connect- 
ed matrix) graph. Three partitions rl, 7r2. and m3 are triorthogonal if 
%-I * R2 13 = 0. 
The next theorem characterizes cube graphs. 
Theorem 3 (Cook). The following are equivalent: 
( 1) G is a cube graph. 
(2) There exist three triorthogonal M-partitions, M,, Ml. and M 3, 
of V(G) containing each line of G exactly twice and Mt l Ml, C + /. Is a 
K-partition af V(G). 
(3) There exist three mutually orthogonal K-partitions. uI, r2. and 
13, of V(G) containing all the lines of G and (Ui+ St)*(Ui+ ft) = 
rl+(ztr/.rk)for 15 i. j. k< 3. 
53. Ihe main theorem 133 
‘Ibe proof of the main theorem requires several lemmas. 
lemma 1. Let L be a lattice with a 0 element and let n, ,x2, and n3 be 
three elements of L with the forrowing properties: 
(I)~i*n,=Ofori#j. I <_i. j<3. 
(2)(~i++~)‘(~i++~)=~i+(ff~‘~~)fOrI <i,j,k<_3. 
Then 
Proof. The proof will be by cases. 
Case l.i=jori=k. 
(~i*‘~)+(~i’U~)“~i==i’(~~++~). 
Case Zj=k, 
(q’~j)+(~i”~)=~i*Uj=“i’(nj+nk). 
Case3,i)j+k+i. 
O=(Wi l Irj)+(Ti'Irk) 
= ((Wi ' S/)+lVi)* ((Xi ' Irj)+W&) (1) 
= ((mi+rf)' (ri+mj))' ((nk +ri)' (nk +rri)) (2) 
=(~~=(~~++j)).((~i++~)'(lli+~&)) commutative laws 
~+i’((~~++~)‘(~/+~~)) absorption laws 
=(*,*(Wd++fi;t))*(W,++k) associative laws 
= rri”’ (Ri + rr~ ) absorption laws 
In the following assume that G is a cubic lattice graph with character- 
istic )I, 
LUIMW 2 The set of n points of C with two common coordinates form 
a clique. 
Roof.lfn=I,thelemmaistrue.Supposethattc,,....u,.n> l,have 
tin same two coordinates in commou. Clearly iirese n points form a 
complete s&graph of G. Any point u adjacent to these n points must 
also have these same two coordinates. Therefore the n points must form 
a clique. 
Lemn~ 3. tttr ul , ~2, and u3 be rhree K-partitions of V(G) H nose blocks 
cvmist of the p&ts of G that have the same first and second, first and 
134 C.R. Cook. Fiuther cbamcteriz~tiom o/cubic btricegmphs 
third, and second and third coordinates, respectively. Then the set of 
points in each block of r1 + r2, ‘A, + n3, and r2 + t3 have the same 
first, second, and third coordinates, respectively. 
Proof. Let u and u be two points in a block of rrl + ~2. From the defi- 
nitio.1 of I, + 7r2, this implies the existence of a sequence u = uo, . . . . up = 
U, where Ui_ 1 sUi(II)OrZl~_I aUi(+2)fori= l,...,p.Thenfori= 
1 , . . ..p. ut_, and Ui have either the same first and second or first and 
third coordi*ates. Hence u and u have the same first coordinate. 
By an analogous argument, the points in each block of rrI +f3 and 
z2 + 7r3 have the same second and third coordinates. 
Theorem 4 (Characterization of cubic lattice graphs). The following 
are equivalent : 
( 1) G is a cubic lattice graph with chamcteristic n. 
(2) There exist three triorthogonal L2-partitions of V(G), M,, M2. 
and M, , such that each Mi contains n blocks of order nl, the three parti- 
tions contain every line of G exactly twice, and Mt l Mi, i * j. is a K-prti- 
tion of V(G). 
(3) There exist three mutually orthogonal K-partitions of V(G), rl, 
t2. and R>, containing all the lines of G, 9 contains na blocks of order 
n and 
0% tIri+Ti)‘(Wi+Uk) ‘Uf+(Ui’ l);t)for 1 5 i. j. k <_ 3. 
Roof. (I? implies (3). Let tl , s2, and 13 be three K-partitions of V(G) 
whose blocks consist of the points of G that agree on the first and 
second, first and third, and second and third coordinates, respectively. 
Clearly nl , x2, and w3 are mutually orthogonal and contain all the lines 
of G. Also each K-partition contains n2 blocks of order n. In fact the 
subgraph induced by each block of Ui is a clique by Lemma 2. 
All that remains is to show that wI, q , and II) satisfy property (D). 
Ifi=jori=k. then 
If j = k, then 
(I3.77te main theorem 
If i # j # k # i. then since rrr , r2, and n3 are mutually orthogonal, 
Assume without loss of generality that (I, + 7r2)*(7r1 + n3) 2 rrl . By 
Lemma 3 the points in each block of rrl + u2 have the same first coordi- 
nate and the points in each block of rr, + n3 have the same second co- 
ordinate. Hence (I, + rr2 )*(t, t r3 ) = tl . 
(3) implies (2). Let rr,, r2, and 7r3 be three K-partitions of V(G) 
satisfying the stated conditions. Define three partitions of V(G). M, , 
M2, and M,, by tl + t2, xl + s3, and rr2 + rt3, respectively. 
First we #ill show that each block Of Mi, i = 1,2,3, is of order u2. 
Let hfi = R,. + wk. Since Uj and rrk are orthogonal and contain n2 blocks 
of order n, each block of Mi must be of order n2 or greater. Suppose a 
b;o,-k of one of the Mi’S, say M, is of order greater than n2. This im- 
plies that M, contains m < n blocks and that M, - M2 contains mq < n2 
blocks where M, contains q <_ n blocks. But this contradicts property 
(D) as 
and wI contain* n* blocks of order 11. 
To show that the lines contained in Mi = m,- + uk are the lines con- 
tained in either wj or rr,, suppose that Mi contains a line not contained 
in either q or +k. Then this line must be contained in rrP, j # p # k. 
i.e.. $8 (q + 4 ) > 0. But this contradicts Lemma 1 as 
Cleariy Mt. ~442, and M3 contain each line of C exactly twice. 
We will use Theorem 2 to show that MI = ui + m& is an L2-partition. 
From the ptcceding, each block X of M, is of order tt2 and Mi contains 
the lines contained in either uI or wt. The n blocks of order n of Rj 
mm4 9 &ch hare a nonempty intersection with X are orthogonal K- 
pa#WiOllaof y’(O). Hence by ‘Theorem 2 the subgraph induced by each 
BkDckotMi b8nt* g@L 
136 CR. Cook, Further chrrmcteritorions f cubic btticegmphs 
It follows almost immediately from (D) that Mt. M,. and M3 are 
triorthogonal: 
M,*M,*M3 =(rl+n,)*(s,$V))"(82+13) 
=(nl+r2)'(a,+r~~(a,+Ij)'(~2+13) 
=((n,in2)'(R~+It))"((tj+Al)o(k3+~~)) 
=(;il+(12’~3))‘(ffj+(R1*~2)) 
=n I l I3 
= 0: 
From (D) and the fact that rrI, 7r2, and rr3 are orthogonal K-parti- 
tions, it follows that Mio Mp i #j. is a K-partition of V(G). 
(2) implies (1). Let M, , M,, and M, be three L2-partitions satisfying 
the stated conditions, where M, = iU,, . ..% V, 1. :M2 = IV,, . . . . V,, 1, and 
A!!, = !W,, . . . . w,] and /r/I = Iv11 = IWkI=n2. DefhleamapphlgffKIm 
V(G) into a cubic lattice graph H with characteristic n by f(u) = (i, j. k) 
if and only if u E Vi n vi n wk. Clearly f is 1 - 1 and onto. 
We must show that f preserves adjacency. Let uu be a line. Since M, , 
M,, and M3 contain each line of G exactly twice, u 3 u (Mi) and 
u 3 u (Mj), f(u) and f(u) must agree on two coordinates and hence are 
adjacent. Conversely, if f(u) is adjacent to f(v). thenf(u) and f(u) must 
agree on two coordinates. This implies that u and u are in the same block 
of Mi* Mi, i # i. But every biock Of Mi* Mj, i #i, induces a complete sub- 
graph in G and hence u is adjacent o u. 
Therefore G r H. 
One might conjecture that property (D) is supertluous. But Fig. 1 shows 
that this is not the case. 
The three partitions. rrt , lr2, and u3, are mutually orthogonal, contain 
each line of G, and each partition contains 22 blocks of order 2. But 
these partitions do not satisfy (D) as 
(I, +“2)‘(“, + f3) = WI + A) > ICI = WI * (f* l w3). 
The graph G is not a cubic lattice graph as points I‘ : and u3, and u6 and 
v7 do not satisfy property (B3) (see Introduction). 
54. Conjecture 137 
G = 
“4 
“8 
Fig. 
“2 
“5 
1. 
4. Conjecture 
The dh interchange graph I,(G) of G is a graph whose points are the 
complete subgraphs & or+ n + 1 of G and two points of I,(G) are 
adjacent if and only if the corresponding K,,, ‘s have a K,, in common. 
The line gmph L(C) of G is I, (G). An m-partite graph G is a graph whose 
points can be partitioned into m subsets V,, . . . . V, such that every line 
joins Vt with Vi* i # j. A bigraph is a 2-partite graph. A complete m-par- 
tite graph contains every line joining Vi with Vi- We write G = KP, ,...,p,n 
if Vi haspi pointsfori= l,...,m. 
It follcws immediately that an L2 graph is the line graph of K,, “. We 
have a similar result for cubic lattice graphs. 
llwowm 5. The cubk lattice graph with characteristic n is isomorphic 
to 12 (K,,&” )* 
Roof. Let the points of Kn,n,n beLlU VU W,whereiJ= (IC~,...U,~, 
Y = (u, ( ..‘) u,,I,and W= (w,..... W, I. Then ui, t.Ji, wk form a K, for 
I <_ i. j, k 5 n. I,et the point Zi4.k denote this K3 in 12(Kn,,,,). Let G be 
the cubic lattice Faph with characteristic n. Then the mappingf from 
I’(/~(&,, )) into V(G) defined by f(~~~,~) = (i, j, k) is clearly I-1, onto, 
and preserves adjacency. Hence I2 ( K,,n,n ) I G. 
In (3). Criinbaum mentions that for m > 3 interchange graphs of 
complete nr-partite graphs do not seem to have been investigated. The- 
orem 4 seems capable of being extended to these graphs. Let L, (n) 
denote the (n+ 1 P interchange graph of the complete m-partite graph 
138 C. R. Cook, Further charactuizations of cubic latticegmphs 
K TheI, an L, graph is denoted L*(n) and a cubic lattice graph 
w%‘&racterMic n is denoted L 3(n). That is, the points of L ,,, (n) 
can be identified with the nm ordered m-tuples on n symbols such that 
two points are adjacent if and only if the corresponding m-tuples have 
m- 1 coordinates in common. A partition A of V(C) is an L,,, (n)_purti- 
rbn if the subgraph induced by each block of u is an L, (n) graph. 
Conjecture. The following are equivalent. 
(1) G is the interchange graph of the complete m-partite graph 
K n.....n for m ' 3* 
(2) There exist m m-orthogonal L,,,_l (n>partitior,s Ml, .., M, of 
V(G) such that each Mj contains n blocks of order nm-*, M,, . . . . M,_, , 
and M,,, contain every line of G exactly m- 1 times, and the subgraph 
induced by each block of the partition formed by the product of m- I 
distinct Mi'S is K,. : 
(3) There exist m mutually orthogonalSK-partidons qlt . . . . q,, of 
V(G) containing all the lines of G, each partition contains tam-1 blocks 
of order n and the lattice generated by these partitions is distributive. 
References 
(I] KIII~, The uniqpeness of the cubic latiicc. g;?h. J. Combinatorial Theory 6 (1969) 
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131 B. Griinbaum. Incidence patterns of graphs and complexes, in: C. Chartrand and 8.F. 
Kapoor. eds., The many facets of graph theory (Spriryr. New York, 1969) 1 IS- 128. 
141 S. Hedctniemi, Graphs of (0. I)-matrices, in: M. Capobianco, J.B. Fnnchen and M. Krolii. 
eds., Recent trends In graph theory (Springer, New York, 1970) 157-172. 
ISI R. Laskar, A characterization of cubic lattice graphs, J. Combinatorial Theory 3 (I 967) 
3z;-401. 
I61 R. Laskar, Eigenvalues SJf the adjacency matrix of cubic lattice graph& Pacific. 1. Math. 
29 (1969) 623-629. 
1’1 J.W. M@on. On the line graph of the complete b&mph. Ann. Math. Statist. 34 (1%)) 664- 
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Further characterizations of cubic lattice graphs 

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Further characterizations of cubic lattice graphs

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