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soliton923
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2楼2011-06-20 16:52:33
racoon01
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3楼2011-06-20 17:01:10
racoon01
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4楼2011-06-20 17:54:38
racoon01
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5楼2011-06-20 20:14:13
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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 |
6楼2011-06-21 09:45:07
racoon01
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7楼2011-06-21 10:53:36
8楼2011-06-22 10:00:12
lianboyong
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9楼2011-09-30 21:30:09
