Vertex arboricity of graphs embedded in a surface of non-negative Euler characteristic

2020 ◽  
Vol 12 (06) ◽  
pp. 2050080
Author(s):  
Wenshun Teng ◽  
Huijuan Wang
Keyword(s):  
Euler Characteristic ◽  
Planar Graphs ◽  
Quadrilateral Mesh ◽  
Color Class ◽  
Vertex Arboricity ◽  
Minimum Number ◽  
Acyclic Subgraph

The vertex arboricity [Formula: see text] of a graph [Formula: see text] is the minimum number of colors the vertices of the graph [Formula: see text] can be colored so that every color class induces an acyclic subgraph of [Formula: see text]. There are many results on the vertex arboricity of planar graphs. In this paper, we replace planar graphs with graphs which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text]. We prove that for the graph [Formula: see text] which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text] if no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, or no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, then [Formula: see text] in addition to the [Formula: see text]-regular quadrilateral mesh.

LIST POINT ARBORICITY OF GRAPHS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250027 ◽  
Author(s):  
NINI XUE ◽  
BAOYINDURENG WU
Keyword(s):  
Planar Graph ◽  
Complete Graph ◽  
Upper Bound ◽  
Discrete Math ◽  
Odd Order ◽  
Color Class ◽  
Linear Arboricity ◽  
Minimum Number ◽  
Acyclic Subgraph

Let G be a graph. The point arboricity of G, denoted by ρ(G), is the minimum number of colors that can be used to color the vertices of G so that each color class induces an acyclic subgraph of G. Borodin et al. (Discrete Math.214 (2000) 101–112) first introduced the list point arboricity of G, denoted by ρl(G). We prove that for any graph G, [Formula: see text], where deg (G) denotes the degeneracy of G, that is, the minimum number k such that δ(H) ≤ k for any subgraph H of G. Using this upper bound, we show that ρl(G) ≤ 3 for any planar graph G. In particular, if either G is K4-minor free, or for an integer k ∈ {3, 4, 5, 6}, G is planar and does not contain k-cycles, then ρl(G) ≤ 2. For any graph G of order n, [Formula: see text]. In addition, we provide a new proof of a theorem of Borodin et al., which states that if G is neither a complete graph of odd order nor a cycle then [Formula: see text]. Finally, we show that la (G) = lla (G) = 2 if G is 3-regular, and la (G) = lla (G) = 3 if G is 4-regular, where la (G) is the linear arboricity of G and lla (G) is list linear arboricity of G which is introduced recently by An and Wu.

Strong Equitable Vertex Arboricity in Cartesian Product Networks

2021 ◽  
pp. 2150010
Author(s):  
Zhiwei Guo ◽  
Yaping Mao ◽  
Nan Jia ◽  
He Li
Keyword(s):  
Maximum Degree ◽  
Cartesian Product ◽  
Color Class ◽  
Vertex Arboricity

An equitable [Formula: see text]-tree-coloring of a graph [Formula: see text] is defined as a [Formula: see text]-coloring of vertices of [Formula: see text] such that each component of the subgraph induced by each color class is a tree of maximum degree at most [Formula: see text], and the sizes of any two color classes differ by at most one. The strong equitable vertex [Formula: see text]-arboricity of a graph [Formula: see text] refers to the smallest integer [Formula: see text] such that [Formula: see text] has an equitable [Formula: see text]-tree-coloring for every [Formula: see text]. In this paper, we investigate the Cartesian product with respect to the strong equitable vertex [Formula: see text]-arboricity, and demonstrate the usefulness of the proposed constructions by applying them to some instances of product networks.

Vertex arboricity of planar graphs without intersecting 5-cycles

2017 ◽  
Vol 35 (2) ◽  
pp. 365-372 ◽  
Author(s):  
Hua Cai ◽  
Jianliang Wu ◽  
Lin Sun
Keyword(s):  
Planar Graphs ◽  
Vertex Arboricity

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

1969 ◽  
Vol 21 ◽  
pp. 992-1000 ◽  
Author(s):  
L. W. Beineke
Keyword(s):  
Projective Plane ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Graphs ◽  
Minimum Number ◽  
Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

2020 ◽  
Vol 12 (04) ◽  
pp. 2050035
Author(s):  
Danjun Huang ◽  
Xiaoxiu Zhang ◽  
Weifan Wang ◽  
Stephen Finbow
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring ◽  
Discrete Math ◽  
Adjacent Vertex ◽  
Chromatic Index ◽  
Proper Edge Coloring ◽  
Minimum Number

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

A note of vertex arboricity of planar graphs without 4-cycles intersecting with 6-cycles

2020 ◽  
Vol 836 ◽  
pp. 53-58
Author(s):  
Xuyang Cui ◽  
Wenshun Teng ◽  
Xing Liu ◽  
Huijuan Wang
Keyword(s):  
Planar Graphs ◽  
Vertex Arboricity

On strict strong coloring of graphs

2020 ◽  
pp. 2150040
Author(s):  
A. Mohammed Abid ◽  
T. R. Ramesh Rao
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Color Class ◽  
Minimum Number ◽  
Strong Chromatic Number

A strict strong coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which every vertex of the graph is adjacent to every vertex of some color class. The minimum number of colors required for a strict strong coloring of [Formula: see text] is called the strict strong chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we characterize the results on strict strong coloring of Mycielskian graphs and iterated Mycielskian graphs.

scholarly journals Introduction to dominated edge chromatic number of a graph

2021 ◽  
Vol 41 (2) ◽  
pp. 245-257
Author(s):  
Mohammad R. Piri ◽  
Saeid Alikhani
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Color Class ◽  
Proper Edge Coloring ◽  
Minimum Number

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph \(G\), is a proper edge coloring of \(G\) such that each color class is dominated by at least one edge of \(G\). The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by \(\chi_{dom}^{\prime}(G)\). We obtain some properties of \(\chi_{dom}^{\prime}(G)\) and compute it for specific graphs. Also examine the effects on \(\chi_{dom}^{\prime}(G)\), when \(G\) is modified by operations on vertex and edge of \(G\). Finally, we consider the \(k\)-subdivision of \(G\) and study the dominated edge chromatic number of these kind of graphs.

scholarly journals Threshold Functions for the Bipartite Turán Property

10.37236/1303 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Anant P. Godbole ◽  
Ben Lamorte ◽  
Erik Jonathan Sandquist
Keyword(s):  
Bipartite Graph ◽  
Exponential Inequalities ◽  
Threshold Functions ◽  
Number Of Zeros ◽  
Bipartite Subgraph ◽  
Color Class ◽  
Turán Number ◽  
Minimum Number ◽  
Complete Bipartite Subgraph ◽  
Asymptotic Probability

Let $G_2(n)$ denote a bipartite graph with $n$ vertices in each color class, and let $z(n,t)$ be the bipartite Turán number, representing the maximum possible number of edges in $G_2(n)$ if it does not contain a copy of the complete bipartite subgraph $K(t,t)$. It is then clear that $\zeta(n,t)=n^2-z(n,t)$ denotes the minimum number of zeros in an $n\times n$ zero-one matrix that does not contain a $t\times t$ submatrix consisting of all ones. We are interested in the behaviour of $z(n,t)$ when both $t$ and $n$ go to infinity. The case $2\le t\ll n^{1/5}$ has been treated elsewhere; here we use a different method to consider the overlapping case $\log n\ll t\ll n^{1/3}$. Fill an $n \times n$ matrix randomly with $z$ ones and $\zeta=n^2-z$ zeros. Then, we prove that the asymptotic probability that there are no $t \times t$ submatrices with all ones is zero or one, according as $z\ge(t/ne)^{2/t}\exp\{a_n/t^2\}$ or $z\le(t/ne)^{2/t}\exp\{(\log t-b_n)/t^2\}$, where $a_n$ tends to infinity at a specified rate, and $b_n\to\infty$ is arbitrary. The proof employs the extended Janson exponential inequalities.

The Strong Perfect Graph Conjecture for Planar Graphs

1973 ◽  
Vol 25 (1) ◽  
pp. 103-114 ◽  
Author(s):  
Alan Tucker
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Independent Set ◽  
Independent Sets ◽  
Clique Number ◽  
Perfect Graph ◽  
Stability Number ◽  
Partition Number ◽  
Minimum Number ◽  
The Stability

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).

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