Dear Referee,

Many thanks for your attention to our paper, careful reading and useful remarks.
Almost all your suggestions are accepted, please see below.

All the best,
Anna Ivanova



++++++++++++++++++++++++++++++++
The weight of 3-paths in sparse plane graphs
Notes to authors and editor
April 19,2015

Dear authors and editor,

in this paper, the authors give some new results on the weight of 3-paths in sparse
plane graphs. They prove precise upper bounds for the minimum weight of a path on
three vertices in plane graphs with minimum degree 2 and girth from 5 to 7.
The main result is a classical application of the discharging method. The results are
new and correct. The paper is well written and I recommend publication in The Electronic
Journal of Combinatorics once the authors take into consideration the following
remarks.

Page 1; Line 16 I would prefer to give the definitions of the
degree of a vertex and of a face before the definition of
\delta(G).
----------------------
Many thanks, done.

Page 2; Line -1 replace "ax_iy_yb" with "ax_iy_ib".
Page 3; Line 1 I suggest to give Theorem 2 in the version from the paper [16] (with
the precise description of types of 3-paths).
----------------------
Thanks, both done.

Page 3; Line -3 there is not mentioned any graph for showing
the optimality of type (2; 2;\infty; 2).
----------------------
Thanks. We rewrote this place as follows:
''Forbidding $(2,2,\infty,2)$-paths in Theorem~\ref{main3} is
justified by the already mentioned graph with $w_3=\infty$ and
$g=6$ in which vertices $a$ and $b$ are joined by independent
paths $ax_iy_ib$ with $1\le i\le n$. We note that forbidding
$(2,2,\infty,2)$-paths still allows both $(2,2,\infty)$-paths and
$(2,\infty,2)$-paths. The sharpness of the bound on $w_3$ follows
by putting a 2-vertex on every edge of the icosahedron, which
results in $w_3=2+5+2$ under the absence of
$(2,2,\infty,2)$-paths.''

Page 5; Line -8 replace all "5+" with 5 (in Figure 2 also) - authors suppose that f is
incident with three vertices of degree from 4 to 5, so bigger degrees are
not considered.
----------------------
Many thanks, we replaced both 5+s in line -8 and the rightmost
5+ in the top row in Fig. 2.

Page 5; Line -5 you could replace "f gives at most 4 1\2" with
"f gives at most 4 x 1\2 ".
----------------------
No, here f gives (at most) just 4\frac12 (=2\times 2+\frac12 for a
(2,2,4^+,\ldots)-face. We replace ''4\frac12''by $\frac92$. 

Page 6; Line 23 the notation "P3 \in \partial(f') \
\partial(f)" is not correct in that case, therefore I suggest
to reformulate it or give some definition of it.
----------------------
Done as follows: "... every $P_3$ that lies in $\partial(f')$
but not in $\partial(f)$ satisfies either $P_3=v_1v_3v_4$ or
$P_3=v_3v_1v_{d(f)}$. ..."

Page 7; Figure 3 I suggest to draw all five edges incident
with vertex v in all cases (instead of a digit 5 in cases (a)
and (b)) and (in at least one case, say in (a)) add a labelling
of the vertices for better clarity.
----------------------
Many thanks, done.

Page 7; Line 10 replace "d(v2) \ge 3" with "d(v3) \ge 3".
Page 7; Line 11 replace "(see Fig. 3c)" with "(see Fig.
3(c-d))".
----------------------
Thanks, both done.

Page 7; Line 12 replace "(see Fig. 3c)" with "(see Fig. 2)".
----------------------
Thanks, we added ''and Fig.\ref{f2}''.

Page 8; ref. 16 replace "M.Macekova Describing" with "M.Macekova, Describing", and
"Discrete Mathematics" with "Discrete Math.".
----------------------
Thanks, done.
