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84127546

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soliton923

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soliton;sato-theory;algebre-geometry£»Random-Matrices-Theory; Riemann-Hilbert method
2Â¥2011-06-20 16:52:33
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racoon01

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84127546(½ð±Ò+1): 2011-06-20 17:16:15
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Originally posted by 84127546 at 2011-06-20 15:42:52:
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racoon
3Â¥2011-06-20 17:01:10
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racoon01

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84127546(½ð±Ò+2): ллÄ㣬»¹ÊDz»ÐÐ 2011-06-20 19:20:02
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racoon
4Â¥2011-06-20 17:54:38
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racoon01

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racoon
5Â¥2011-06-20 20:14:13
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84127546

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\magnification=1200 \baselineskip14pt\centerline{\bf Referee Report}
\vskip10pt\leftline{{\bf Title}: Short Cycle Covers of Cubic Graphs}
\leftline{{\bf Authors}: Daniel Kral et al.} \leftline{{\bf
Journal}: SIAM Journal on Discrete Mathematics} \leftline{{\bf File
Number}: SIDMA71746} \vskip10pt

The paper studies the short cycle covers of cubic graphs, and gives
a new upper bound on the total length of a cycle cover of cubic
graphs. For a general bridgeless graph with $m$ edges, $5m/3$ is the
best known upper bound on the total length of a cycle cover, which
was obtained by two different groups of people about 25 years ago.
Since then, no breakthrough was made. However, for a bridgeless
cubic graph, it was Jackson who first showed that the upper bound
can be improved to $64m/39$, and then to $44m/27$ by Fan. After
almost 15 years, the current paper improves the bound to $34m/21$.
This is interesting and significant, and should be published. The
only reservation I have is that the improvement is not obtained by a
new approach or new method. It is just a harder push, through the
same approach as that in ref.[5].

It is known that a bridgeless cubic graph $G$ has a 2-factor $F$
such that $G/F$, the graph obtained by contracting all the edges of
$F$, contains no 3-cuts, and so has a nowhere-zero 4-flow. Let
$cc(G)$ denote the minimum length of a cycle cover of $G$ and let
$d_\ell$ be the number of circuits of length $\ell$ in $F$. Suppose
that $G$ has $m$ edges. The following two inequalities are proved in
ref.[5].
$$cc(G)\leq {19\over 12} m+ {5\over 4} d_2+{5\over 8} d_5+
{3\over 4} d_6 - {1\over 8} d_7.$$
$$cc(G)\leq {5\over 3} m- d_2- {1\over 2} d_5 -d_6 -
{3\over 2} d_7.$$ Adding 5/4 times the second inequality to the
first one yields the upper bound $cc(G)\leq 44m/27$. In these two
inequalities, circuits of lengths more than 7 are not included. As
showed in ref.[5], circuits of length $4t$ $(t\geq2)$ are the main
"trouble". The only contribution of the current paper is the third
part of Lemma 12, which enable one to have an improvement on
covering circuits of length 8 and them improve the first inequality
above. I must say that I am not happy with the way used to overcome
the circuits of length 8 in the paper and have no patience to check
all the 16 cases in the third part of Lemma 12. If the editor can
get someone to check the correctness of the third part of Lemma 12,
and if the paper is revised according to the following comments,
then I think that it would be publishable.

\vskip15pt\noindent {\bf Comment 1}. In the introduction ( top of
page 2 ), it is said that Fulkerson Conjecture implies the Cycle
Double Cover Conjecture. To my knowledge, no one has proved this!
Fulkerson Conjecture gives a perfect matching 2-cover, which yields
a cycle 4-cover, not double cover!

\vskip8pt\noindent {\bf Comment 2}. Section 3 ( pages 4 to 8 )
should be simplified significantly. There is no need to use
"odd-connectedness". What you need here is just the property: the
vertex splitting creates no new 2-cuts, which has been studied well
in the literature.

\vskip8pt\noindent {\bf Comment 3}. There is no need for Lemma 9 (
Rainbow Lemma ). What you need here is the known result: every
bridgeless cubic graph $G$ has a 2-factor $F$ such that $G/F$ has a
nowhere-zero 4-flow. Manipulating this 4-flow is equivalent to
adjusting the 3-edge-coloring used in the proofs.

\vskip8pt\noindent {\bf Comment 4}. Page 13, line 13: notation used
$\to$ notations to be used

\vskip8pt\noindent {\bf Comment 5}. The whole section 5 should be
removed, which is entirely contained in the proof of the inequality
at the bottom of page 138 in ref.[5].

\vskip8pt\noindent {\bf Comment 6}. On pages 17-18, there is no need
to give a full proof of "the second cycle cover". Equation (9) can
be derived from the inequality in Lemma 3.4 of ref.[5] by using
$|V(G^*)|\leq \sum_{i=2}^{11}d_i+{1\over 12}(|F|-\sum_{i=2}^{11}
id_i)$. Note that because of the relaxation on $d_2$, an inequality
weaker than the one in Lemma 3.4 of ref.[5] is enough here, which
does not need the use of Lemma 3.3 in ref.[5].

\vskip15pt In summary, to obtain the result, the authors only need
to get equation (5), which is an improvement on the inequality at
the bottom of page 138 in ref.[5], by using the third part of Lemma
12. In this way, together with the above comments, the paper can be
shortened significantly. If the editor can have someone check the
correctness of the third part of Lemma 12, then such a revised
version should be acceptable.


\end
6Â¥2011-06-21 09:45:07
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racoon01

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ÒýÓûØÌû:
Originally posted by 84127546 at 2011-06-21 09:45:07:
\magnification=1200 \baselineskip14pt\centerline{\bf Referee Report}
\vskip10pt\leftline{{\bf Title}: Short Cycle Covers of Cubic Graphs}
\leftline{{\bf Authors}: Daniel Kral et al.} \leftline{{\ ...

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racoon
7Â¥2011-06-21 10:53:36
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84127546

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ллÄã
8Â¥2011-06-22 10:00:12
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lianboyong

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