The notion of a $\mathcal C$-ultrahomogeneous graph, due to Isaksen et al.,
is adapted for digraphs, and subsequently a strongly connected
$\vec{C}_4$-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise
arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and
no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph
via pencils of ordered lines of the Fano plane in which pencils are replaced by
ordered pencils.Comment: 4 pages, 2 figures, 2 table

Dejter, Italo J.

English

arXiv

AbstractThe notion of a C-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs, and then a strongly connected C→4-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in which pencils are replaced by ordered pencils

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On a C→4-ultrahomogeneous oriented graph 

Discrete Mathematics 310 (2010) 1389–1391
Contents lists available at ScienceDirect
Discrete Mathematics
journal homepage: www.elsevier.com/locate/disc
Note
On a EC4-ultrahomogeneous oriented graph
Italo J. Dejter
University of Puerto Rico, Rio Piedras, PR 00931-3355, Puerto Rico
a r t i c l e i n f o
Article history:
Received 17 August 2009
Received in revised form 2 December 2009
Accepted 10 December 2009
Available online 4 January 2010
Keywords:
Ultrahomogeneous oriented graph
Fano plane
Ordered pencil
a b s t r a c t
The notion of a C-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs,
and then a strongly connected EC4-ultrahomogeneous oriented graph on 168 vertices and
126 pairwise arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and
no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of
ordered lines of the Fano plane in which pencils are replaced by ordered pencils.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
The study of ultrahomogeneous graphs (resp. digraphs) can be traced back to [12,6,11,7], (resp. [5,10,2]). In [9], C-
ultrahomogeneous graphs are defined and subsequently treated when C is a collection of either (a) complete graphs, or
(b) disjoint unions of complete graphs, or (c) complements of those unions. In [3], a {K4, K2,2,2}-ultrahomogeneous graph on
42 vertices, 42 copies ofK4 and 21 copies ofK2,2,2 is given that fastens objects of (a) and (c), namelyK4 andK2,2,2, respectively,
over copies of K2.
In the present note and in [4], the notion of a C-ultrahomogeneous graph is extended as follows: Given a collection C of
(di)graphs closed under isomorphisms, a (di)graph G is C-ultrahomogeneous (or C-UH) if every isomorphism between two
G-induced members of C extends to an automorphism of G. If C = {H} is the isomorphism class of a (di)graph H , such a G
is said to be {H}-UH or H-UH.
In [4], the cubic distance-transitive graphs are shown to be Cg-UH graphs, where Cg is a cycle realizing the girth g .
Moreover, all of them, but for the Petersen, Heawood and Foster graphs, are shown to be ECg-UH digraphs. This allows
the construction of novel C-UH graphs, in a continuation of the work of [3]. In particular, these include a {K4, L(Q3)}-UH
graph on 102 vertices that fastens 102 copies of K4 and 102 copies of the cuboctahedral graph L(Q3) over copies of K3. In its
construction from the Biggs–Smith graph, [1], we use unzipping, powering and zipping back a collection of oriented g-cycles
provided by the initial results. However, the mentioned C-UH graphs are undirected, so they are not proper digraphs.
In this note, a presentation of the Coxeter graph Cox is modified to provide a strongly connected EC4-UH oriented graph
D on 168 vertices, 126 pairwise arc-disjoint 4-cycles, with regular indegree and outdegree 3. In contrast, the construction
of [3] used ordered pencils of unordered lines.
We take the Fano plane F as having point set J7 = Z7 (the cyclic group mod 7) and point-line correspondence
φ(j) = {(j+ 1), (j+ 2), (j+ 4)}, for every j ∈ Z7, in order to color the vertices and edges of Cox as in Fig. 1.
This figure shows that each vertex v of Cox can be considered as an unordered pencil of ordered lines ofF , (brackets and
commas avoided now):
xb1c1, xb2c2, xb0c0, (1)
E-mail address: ijdejter@uprrp.edu.
0012-365X/$ – see front matter© 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.disc.2009.12.008
1390 I.J. Dejter / Discrete Mathematics 310 (2010) 1389–1391
Fig. 1. Coloring the vertices and edges of Coxwith elements of F .
corresponding to the three edges e1, e2, e0 incident to v, respectively, and denoted by [x, b1c1, b2c2, b0c0], where x is the
color of v in the figure, with bi and ci as the colors of the edge ei and the endvertex of ei other than v, for i ∈ {1, 2, 0}.
Moreover, two such vertices are adjacent in Cox if they can be written [x, b1c1, b2c2, b0c0] and [x′, b′1c ′1, b′2c ′2, b′0c ′0]
(perhaps bymeans of a permutation of the entries bici) in such a way that {bi, ci}∩ {b′i, c ′i } is constituted by just one element
di, for each i ∈ {1, 2, 0}, and the resulting triple d1d2d0 is a line of F .
2. Presentation of a EC4-UH digraph
Consider the oriented graph D whose vertices are the ordered pencils of ordered lines of F , as in (1) above. Each
such vertex will be denoted (x, b1c1, b2c2, b0c0), where b1b2b0 is a line of F . An arc between two vertices of D, say from
(x, b1c1, b2c2, b0c0) and (x′, b′1c
′
1, b
′
2c
′
2, b
′
0c
′
0), is established if and only if
x = c ′i , b′i+1 = ci+1, b′i−1 = ci−1, b′i = bi,
x′ = ci, c ′i+1 = bi−1, c ′i−1 = bi+1,
for some, i ∈ {1, 2, 0}. This way, we obtain oriented 4-cycles in D, such as
((0, 26, 54, 31), (6, 20, 43, 15), (0, 26, 31, 54), (6, 20, 15, 43)).
A simplified notation for the vertices (x, yz, uv, pq) of D is yupx. With such a notation, the adjacency sub-list of D departing
from the vertices of the form yup0 is (with rows indicated a, b, c, d, e, f , to be used below):
1240:1653,3256,3645;
1420:1563,3465,3526;
2350:2146,6341,6156;
2530:2416,6514,6436;
3460:3521,1425,1562;
3640:3251,1652,1245;
1560:1423,3524,3462;
1650:1243,3642,3254;
2140:2356,6153,6345;
2410:2536,6435,6513;
3250:3641,1246,1651;
3520:3461,1564,1421;
4360:4125,5321,5162;
4630:4215,5612,5231;
5160:5324,4123,4362;
5610:5234,4632,4213;
4120:4365,5163,5326;
4210:4635,5236,5613;
5230:5614,4216,4634;
5320:5164,4361,4124;
6340:6152,2351,2145;
6430:6512,2415,2531;
6150:6342,2143,2354;
6510:6432,2534,2413.
From this sub-list, the adjacency list of D, for its 168 = 24× 7 vertices, is obtained via translations mod 7. Let us represent
each vertex yup0 of D by means of a symbol ji, where i = a, b, c, d, e, f stands for the successive rows of the table above and
j = φ−1(yup) ∈ {0, 1, 2, 4}. These symbols ji are assigned to the lines yup avoiding 0 ∈ F , and thus to the vertices yup0, as
follows:
ji j=0 j=1 j=2 j=4
i=a
i=b
124
142
235
253
346
364
156
165
i=c
i=d
214
241
325
352
436
463
516
561
i=e
i=f
412
421
523
532
634
643
615
651
With these symbols adopted, the quotient graph D/Z7 can be considered as a voltage graph with group Z7 and derived
graph D, [8], admitting a split representation into the three connected digraphs of Fig. 2, whose vertices are indicated by
the symbols ji of their representatives yup0, and in which: (1) the 18 oriented 4-cycles that are shown are interpreted all
with counterclockwise orientation; (2) For each i ∈ {a, . . . , f }, the three vertices indicated by 0i represent just one vertex of
D/Z7, so theymust be identified; (3) the leftmost arc in each one of the three connected digraphsmust be identifiedwith the
corresponding rightmost arc by parallel translation; (4) if an arc Ee of D/Z7 has voltage ν ∈ Z7, initial vertex ji and terminal
vertex j′i′ , then a representative yupµ of ji initiates an arc in D that covers Ee and has terminal vertex y′u′p′(ν+µ), where yup0
and y′u′p′0 are represented respectively by ji and j
′
i′ .
All the oriented 4-cycles of D are obtained by uniform translations mod 7 from these 18 oriented 4-cycles. Thus, there
are only 126 = 7× 18 oriented 4-cycles of D. Our construction of D shows that the following statement holds.
I.J. Dejter / Discrete Mathematics 310 (2010) 1389–1391 1391
Fig. 2. Split representation of the quotient graph D/Z7 .
Theorem 1. The oriented graph D is a strongly connected EC4-UH digraph on 168 vertices, 126 pairwise disjoint oriented 4-cycles,
with regular indegree and outdegree both equal to 3 and no circuits of lengths 2 and 3. 
References
[1] N.L. Biggs, D.H. Smith, On trivalent graphs, Bull. Lond. Math. Soc. 3 (1971) 155–158.
[2] G.L. Cherlin, The Classification of Countable Homogeneous Directed Graphs and Countable Homogeneous n-Tournaments, Memoirs Amer. Math. Soc,
vol. 131, number 612, Providence, RI, 1988.
[3] I.J. Dejter, On a {K4, K2,2,2}-ultrahomogeneous graph, Australas. J. Combin. 44 (2009) 63–76.
[4] I.J. Dejter, On certain C-ultrahomogeneous graphs obtained from cubic distance-transitive graphs, preprint, 2009.
[5] R. Fraïssé, Sur l’extension aux relations de quelques proprietés des ordres, Ann. Sci. Éc. Norm. Supér. 71 (1954) 363–388.
[6] A. Gardiner, Homogeneous graphs, J. Combin. Theory Ser. B 20 (1976) 94–102.
[7] Ja.Ju. Gol’fand, M.H. Klin, On k-homogeneous graphs (in Russian);
Ja.Ju. Gol’fand, M.H. Klin, On k-homogeneous graphs, in: Algorithmic Studies in Combinatorics, vol. 186, Nauka, Moscow, 1978, pp. 76–85 (in Russian).
[8] J. Gross, T.W. Tucker, Topological Graph Theory, Wiley, New York, 1987.
[9] D.C. Isaksen, C. Jankowski, S. Proctor, On K∗-ultrahomogeneous graphs, Ars Combin. LXXXII (2007) 83–96.
[10] A.H. Lachlan, R. Woodrow, Countable ultrahomogeneous undirected graphs, Trans. Amer. Math. Soc. 262 (1980) 51–94.
[11] C. Ronse, On homogeneous graphs, J. Lond. Math. Soc. (2) 17 (1978) 375–379.
[12] J. Sheehan, Smoothly embeddable subgraphs, J. Lond. Math. Soc. (2) 9 (1974) 212–218.


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