On the Pseudoachromatic Index of the Complete Graph III
On the Pseudoachromatic Index of the Complete Graph III
AraujoPardo, M.; MontellanoBallesteros, Juan; RubioMontiel, Christian; Strausz, Ricardo
20180122 00:00:00
An edge colouring of a graph G is complete if for any distinct colours
$$c_1$$
c
1
and
$$c_2$$
c
2
one can find in G adjacent edges coloured with
$$c_1$$
c
1
and
$$c_2$$
c
2
, respectively. The pseudoachromatic index of G is the maximum number of colours in a complete edge colouring of G. Let
$$\psi (n)$$
ψ
(
n
)
denote the pseudoachromatic index of
$$K_n$$
K
n
. In the paper we proved that if
$$ x\ge 2 $$
x
≥
2
is an integer and
$$n\in \{4x^2x,\dots ,4x^2+3x3\}$$
n
∈
{
4
x
2

x
,
⋯
,
4
x
2
+
3
x

3
}
, then
$$\psi (n) \le 2x(nx1)$$
ψ
(
n
)
≤
2
x
(
n

x

1
)
. Let q be an even integer and let
$$ m_a=(q+1)^2a $$
m
a
=
(
q
+
1
)
2

a
. If there is a projective plane of order q, a complete edge colouring of
$$K_{m_a}$$
K
m
a
with
$$(m_aa)q$$
(
m
a

a
)
q
colours,
$$ a\in \{1,0,\dots ,\frac{q}{2}+1\}$$
a
∈
{

1
,
0
,
⋯
,
q
2
+
1
}
, is presented. The main result states that if
$$q\ge 4$$
q
≥
4
is an integer power of 2, then
$$\psi (m_a)=(m_aa)q$$
ψ
(
m
a
)
=
(
m
a

a
)
q
for any
$$ a\in \{1,0,\dots ,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil 1 \} .$$
a
∈
{

1
,
0
,
⋯
,
1
+
4
q
+
9
2

1
}
.
http://www.deepdyve.com/assets/images/DeepDyveLogolg.pngGraphs and CombinatoricsSpringer Journalshttp://www.deepdyve.com/lp/springerjournals/onthepseudoachromaticindexofthecompletegraphiiide1YmPUWWp
On the Pseudoachromatic Index of the Complete Graph III
An edge colouring of a graph G is complete if for any distinct colours
$$c_1$$
c
1
and
$$c_2$$
c
2
one can find in G adjacent edges coloured with
$$c_1$$
c
1
and
$$c_2$$
c
2
, respectively. The pseudoachromatic index of G is the maximum number of colours in a complete edge colouring of G. Let
$$\psi (n)$$
ψ
(
n
)
denote the pseudoachromatic index of
$$K_n$$
K
n
. In the paper we proved that if
$$ x\ge 2 $$
x
≥
2
is an integer and
$$n\in \{4x^2x,\dots ,4x^2+3x3\}$$
n
∈
{
4
x
2

x
,
⋯
,
4
x
2
+
3
x

3
}
, then
$$\psi (n) \le 2x(nx1)$$
ψ
(
n
)
≤
2
x
(
n

x

1
)
. Let q be an even integer and let
$$ m_a=(q+1)^2a $$
m
a
=
(
q
+
1
)
2

a
. If there is a projective plane of order q, a complete edge colouring of
$$K_{m_a}$$
K
m
a
with
$$(m_aa)q$$
(
m
a

a
)
q
colours,
$$ a\in \{1,0,\dots ,\frac{q}{2}+1\}$$
a
∈
{

1
,
0
,
⋯
,
q
2
+
1
}
, is presented. The main result states that if
$$q\ge 4$$
q
≥
4
is an integer power of 2, then
$$\psi (m_a)=(m_aa)q$$
ψ
(
m
a
)
=
(
m
a

a
)
q
for any
$$ a\in \{1,0,\dots ,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil 1 \} .$$
a
∈
{

1
,
0
,
⋯
,
1
+
4
q
+
9
2

1
}
.
Journal
Graphs and Combinatorics
– Springer Journals
Published: Jan 22, 2018
Recommended Articles
Loading...
References
On the pseudoachromatic index of the complete graph
AraujoPardo, G; MontellanoBallesteros, JJ; Strausz, R
On the pseudoachromatic index of the complete graph II
AraujoPardo, G; MontellanoBallesteros, JJ; RubioMontiel, C; Strausz, R
On complete colorings of graphs. Recent Advances in Graph Theory (Proc. Second Czechoslovak Sympos., Prague, 1974)
Bories, F; Jolivet, JL
Graph Theory with Applications
Bondy, A; Murty, R
Complete and pseudocomplete colourings of a graph
Bosák, J; Nešetřil, J
Indice achromatique des graphes multiparti complets et réguliers
Bouchet, A
Further results on the achromatic number
Geller, D; Kronk, H
Bounds on the chromatic and achromatic numbers of complementary graphs. Rec. Prog. Combin., (Proc. Third Waterloo Conf. on Combin., Waterloo, 1968)
Gupta, R
Graph Theory
Harary, F
Achromatic index of $K_{12}$
Horňák, M
On the achromatic index of $K_{q^2+q}$ for a prime $q$
Horňák, M; Pčola, Š; Woźniak, M
On the edge achromatic number of complete graphs
Jamison, RE
On the achromatic index of $K_{12}$
Jamison, RE
Some improved lower bounds for the edge achromatic number of a complete graph. Graphs Combin. Algorit. and App. (Krishnankoil, 2004)
Sane, SS
The edge achromatic number of small complete graphs
Turner, CM; Rowley, R; Jamison, RE; Laskar, R
You’re reading a free preview. Subscribe to read the entire article.
“Hi guys, I cannot tell you how much I love this resource. Incredible. I really believe you've hit the nail on the head with this site in regards to solving the researchpurchase issue.”
Daniel C.
“Whoa! It’s like Spotify but for academic articles.”
@Phil_Robichaud
“I must say, @deepdyve is a fabulous solution to the independent researcher's problem of #access to #information.”
@deepthiw
“My last article couldn't be possible without the platform @deepdyve that makes journal papers cheaper.”
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Subscribe to Journal Email Alerts
To subscribe to email alerts, please log in first, or sign up for a DeepDyve account if you don’t already have one.
Follow a Journal
To get new article updates from a journal on your personalized homepage, please log in first, or sign up for a DeepDyve account if you don’t already have one.
Our policy towards the use of cookies
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.