Let G be a simple, connected and undirected graph that has a set of vertex and edge. The degree of v ∈ V(G) is denoted by d(v). The maximum and minimum degree of G respectively are Δ(G) and δ(G). The ...r-dynamic color of the graph G is calculated as a map c from V to a color set such that if u, v ∈ V(G) is adjacent, then c(u) ≠ c(v), and for each v ∈ V(G), |c(N(v))| ≥ min{r, d(v)}. The number of r-dynamic coloring of G denoted by χr(G) is minimum color k in G. In this paper, we have obtained the r-dynamic vertex coloring of line, middle, total of lobster graph ℒn(2, 1).
Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from ℤk and that induces a proper vertex coloring ...on G where the color of a vertex v is the sum (in ℤk) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by χt′(G). In this paper, we determine the twin chromatic indices of circulant graphs Cn(1,n2), and some generalized Petersen graphs such as GP(3s, k), GP(m, 2), and GP(4s, l) where n ≥ 6 and n ≡ 0 (mod 4), s ≥ 1, k ≢ 0 (mod 3), m ≥ 3 and m ∉ {4, 5}, and l is odd. Moreover, we provide some sufficient conditions for a connected graph with maximum degree 3 to have twin chromatic index greater than 3.
Let graph (p, q) be a simple graph, if there is a positive integer k(1 ≤ k ≤ |E|) and the mapping f: E(G) → {1, 2, . . . , k}, such that for any edge ð® ∈ E(G) , when d(u) = d(v), S(u) = S(v) , ...where S(u) = ð'-∈ ( ) (ð'-) . ( ) represents the degree of point , then f is the coloring of Adjacent Vertex Sum Reducible Edge of G. The maximum value k is called the Adjacent Vertex Sum Reducible Edge chromatic number of graph G, denoted as ð'-Avsr(G). Based on the existing graph coloring concepts and reducible concepts, this paper proposes a new concept of adjacent vertex sum reducible edge coloring combined with practical problems and designs a new type of adjacent vertex sum reducible edge coloring algorithm.
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Abstract
Let c : V(G) → N be a vertex coloring of a simple, connected graph G. For a vertex v of G, the
color sum
of v, denoted by σ(ν), is the sum of the colors of the neighbors of v. If σ(ω) = σ(ν) ...for any two adjacent vertices u and v of G, then c is called a
sigma coloring
of G. The
sigma chromatic number of G,
denoted by σ(
G
), is the minimum number of colors required in a sigma coloring of G. Let max(c) be the largest color assigned to a vertex of G by a coloring c. The
sigma value
of G, denoted by v(G), is the minimum value of max(c) over all sigma k—colorings c of G for which
ρ
(
G
) =
k.
On the other hand, the
sigma range
of G, denoted by
ρ
(G), is the minimum value of max(c) over all sigma colorings c of G. In this paper, we determine the sigma value and the sigma range of the join of a finite number of even cycles of the same order. In particular, if
n >
4 and n is even, then we will show that
ρ(kC
n
) =
v(kC
n
)
= 2 if and only if (i)
k
≤
n
6
+
1
, whenever n = 0 (mod 4), and (ii)
k
≤
n
−
2
6
+
1
, whenever n = 2 (mod 4).
Let be a proper k-edge coloring of a simple connected graph of order at the latest 3, where the set of colors is. If a proper edge coloring of a graph that induces vertex coloring is a proper vertex ...coloring, then is called a twin edge k-coloring of a graph. the twin chromatic number of a graph is the least value k of colors of a graph, and denote it by. In this paper, we obtains the twin chromatic number of infinite paths and finite paths.
Let Γ be a nontrivial connected graph, c:VΓ⟶ℕ be a vertex colouring of Γ, and Li be the colouring classes that resulted, where i=1,2,…,k. A metric colour code for a vertex a of a graph Γ is ...ca=da,L1,da,L2,…,da,Ln, where da,Li is the minimum distance between vertex a and vertex b in Li. If ca≠cb, for any adjacent vertices a and b of Γ, then c is called a metric colouring of Γ as well as the smallest number k satisfies this definition which is said to be the metric chromatic number of a graph Γ and symbolized μΓ. In this work, we investigated a metric colouring of a graph ΓZn and found the metric chromatic number of this graph, where ΓZn is the zero-divisor graph of ring Zn.
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Abstract
The coloring theory of graphs is an important part of graph theory research. The key problem of the coloring theory of graphs is to determine the coloring number of each kind of coloring. ...Traditional coloring concepts mainly include proper vertex coloring, proper edge coloring, proper total coloring, and so on. In recent years, scholars at home and abroad have put forward some new coloring concepts, such as neighbor vertex distinguishing edge (total) coloring, and neighbor sum distinguishing edge (total) coloring, based on traditional coloring concepts and by adding other constraints. Some valuable results have been obtained, which further enrich the theory of graph coloring.
For a proper
k
-edge coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing
k
-edge coloring of G. For a proper
k
-total coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing
k
-total coloring of G . In this paper, the coloring method and coloring index are determined by the process of induction and deduction and the construction of the dyeing method, and then the rationality of the method is verified by inverse proof and mathematical induction.
If G is a simple graph with the order
n
≥ 5 , and
h
n
= (
H
i
)
i
∈{1,2,…,n}
is a sequence of disjoint simple graphs, where every
H
i
is a simple graph with the order
m
≥ 7 . In this paper, we study the neighbor sum distinguishing edge(total) coloring of the generalized corona product
G○h
n
of
G
and
h
n
. The results are as follows:
(1) If G is a path with order
n
,
h
n
= (
H
i
)
i
∈{1,2,…,n}
is an alternating sequence of path and cycle with order
m
. If
n
is odd, we have
χ
Σ
′
(
G
∘
h
n
)
=
m
+
3
(2) If G is a path with order
n
,
h
n
= (
H
i
)
i
∈{1,2,…,n}
is an alternating sequence of path and cycle with order
m
. If
n
is odd, we have
χ
Σ
′
′
(
G
∘
h
n
)
=
m
+
4
Due to the late development of neighbor sum distinguishing edge (total) coloring of graphs, the related research results are relatively few. By studying the operation graph of a basic simple graph, we can provide the research basis and reference idea for the corresponding coloring of the general graph class. Therefore, it is of theoretical value to study the neighbor sum distinguishing edge (total) coloring problem of generalized corona products of graphs.
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