An equitable k-coloring of a graph G is a proper vertex coloring such that the size of any two color classes differ at most 1. If there is an equitable k-coloring of G, then the graph G is said to be ...equitably k-colorable. A 1-planar graph is a graph that can be embedded in the Euclidean plane such that each edge can be crossed by other edges at most once. An IC-planar graph is a 1-planar graph with distinct end vertices of any two crossings. In this paper, we will prove that every IC-planar graph with girth g≥7 is equitably Δ(G)-colorable, where Δ(G) is the maximum degree of G.
G
has a list
k
-
L
(2, 1)-labeling if for any
k
-list assignment
L
, there exists a coloring
c
:
V
(
G
)
→
⋃
v
∈
V
L
(
v
)
of
G
such that
c
(
v
)
∈
L
(
v
)
for
∀
v
∈
V
(
G
)
and for
∀
u
,
v
∈
V
(
G
)
...,
|
c
(
u
)
-
c
(
v
)
|
≥
2
if
d
(
u
,
v
)
=
1
,
|
c
(
u
)
-
c
(
v
)
|
≥
1
if
d
(
u
,
v
)
=
2
.
λ
2
,
1
l
(
G
)
=
min
{
k
|
G
has a list
k
-
L
(2, 1)-labeling
}
is called the list
L
(2, 1)
-labeling number
of
G
. In this paper, we prove that for planar graph with maximum degree
Δ
≥
5
, girth
g
≥
13
and without adjacent 13-cycles,
λ
2
,
1
l
(
G
)
≤
Δ
+
3
holds. Moreover, the upper bound
Δ
+
3
is tight.
Electrochemiluminescence of the luminol-O(2) system in an electrolyte-free N,N-dimethylformamide (DMF)-dipropylamine (DPA) cosolution is induced by the formation of a carbamate ionic liquid (IL) from ...the reaction between CO(2) and DPA, on the basis of which a facile ECL sensor for measuring atmospheric CO(2) has been developed. This ECL sensing method shows several advantages in the detection of CO(2), such as high safety, high selectivity, wide linear response range, and good sensitivity. The gas sensor was found to have a linear response range from 100 ppm to 100 v/v% and a detection limit of 80 ppm (at signal-to-noise ratio of 3). This is the first reported IL-induced ECL sensor for a gas, thus the principle of this type of sensor and the IL-induced ECL mechanism have been demonstrated in detail.
A graph
G
on
n
vertices is called non-universal if its maximum degree is at most
n
-
2
. In this paper, we give a structural characterization for non-universal maximal planar graphs with diameter ...two. In precise, we find 10 basic graphs, and then generate all 25 non-universal maximal planar graphs with diameter two by adding repeatedly and appropriately 3-vertices to some of these 10 basic graphs. As an application, we show that maximal planar graphs with diameter two are pancyclic except five special graphs.
The adjacent vertex distinguishing edge coloring of a graph
G
is a proper edge coloring of
G
such that the edge coloring set on any pair of adjacent vertices is distinct. The minimum number of colors ...required for an adjacent vertex distinguishing edge coloring of
G
is denoted by
χ
a
′
(
G
)
. It is observed that
χ
a
′
(
G
)
≥
Δ
(
G
)
+
1
when
G
contains two adjacent vertices of degree
Δ
(
G
)
. In this paper, we prove that if
G
is a planar graph without 4-cycles, then
χ
a
′
(
G
)
≤
max
{
9
,
Δ
(
G
)
+
1
}
.
An equitable k-coloring of a graph G is a proper vertex coloring such that the size of any two color classes differ at most 1. If there is an equitable k-coloring of G, then the graph G is said to be ...equitably k-colorable. A 1-planar graph is a graph that can be embedded in the Euclidean plane such that each edge can be crossed by other edges at most once. An IC-planar graph is a 1-planar graph with distinct end vertices of any two crossings. In this paper, we will prove that every IC-planar graph with girth g≥7 is equitably Δ(G)-colorable, where Δ(G) is the maximum degree of G.
The preparation, characterization, fluorescence (FL) and electrochemiluminescence (ECL) of graphite-like carbon nitride nanoflake particles (g-C(3)N(4) NFPs) and nanoflake films (g-C(3)N(4) NFFs) ...have been reported. Highly water-dispersible g-C(3)N(4) NFPs with a height of ~5 to 35 nm and a lateral dimension of ~40 to 220 nm have been extracted from bulk g-C(3)N(4) materials by chemical oxidation. New, stable and defined g-C(3)N(4) NFFs can be easily obtained by drying NFPs on certain hydrophilic substrates such as glass or electrode surfaces. Both g-C(3)N(4) NFPs and g-C(3)N(4) NFFs have good FL activities, i.e. they can give strong blue light (435 nm) emission under UV light (365 nm) excitation. The as-prepared g-C(3)N(4) NFFs on a glassy carbon electrode exhibit strong non-surface state ECL activity in the presence of reductive-oxidative coreactants, including dissolved oxygen (O(2)), hydrogen peroxide (H(2)O(2)) and peroxydisulfate (S(2)O(8)(2-)) and give rise to blue light emission (435 nm), which is the same as the wavelength of FL. The non-surface state ECL mechanisms of the g-C(3)N(4) NFF-coreactant systems have been studied and discussed in detail.
An adjacent vertex distinguishing total coloring of a graph
G
is a proper total coloring of
G
such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed ...for an adjacent vertex distinguishing total coloring of
G
is denoted by
.
In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs
G
with large maximum degree
Δ
by showing that if
Δ
≥14, then
, and
if and only if
G
contains two adjacent vertices of maximum degree.
A graph G is k-choosable if G can be properly colored whenever every vertex has a list of at least k available colors. In this paper, we will proof that if every 5-cycle of toroidal graph G is not ...adjacent simultaneously to 3-cycles and 4-cycles, then G is 4-choosable. This improves a result shown in Xu and Wu (2017), which stated that if every 5-cycle of planar graph G is not adjacent simultaneously to 3-cycles and 4-cycles, then G is 4-choosable.
