Minimum degree condition for proper connection number 2

2019 ◽  
Vol 774 ◽  
pp. 44-50 ◽  
Author(s):  
Fei Huang ◽  
Xueliang Li ◽  
Zhongmei Qin ◽  
Colton Magnant
Keyword(s):  
Minimum Degree ◽  
Connection Number ◽  
Degree Condition ◽  
Proper Connection Number

NOTE ON MINIMUM DEGREE AND PROPER CONNECTION NUMBER

2020 ◽  
pp. 1-5
Author(s):  
YUEYU WU ◽  
YUNQING ZHANG ◽  
YAOJUN CHEN
Keyword(s):  
Minimum Degree ◽  
Connection Number ◽  
Minimum Value ◽  
Proper Connection Number ◽  
Degree Conditions

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a properly coloured path. The proper connection number, $pc(G)$ , of a graph $G$ , is the smallest number of colours that are needed to colour $G$ such that it is properly connected. Let $\unicode[STIX]{x1D6FF}(n)$ denote the minimum value such that $pc(G)=2$ for any 2-connected incomplete graph $G$ of order $n$ with minimum degree at least $\unicode[STIX]{x1D6FF}(n)$ . Brause et al. [‘Minimum degree conditions for the proper connection number of graphs’, Graphs Combin.33 (2017), 833–843] showed that $\unicode[STIX]{x1D6FF}(n)>n/42$ . In this note, we show that $\unicode[STIX]{x1D6FF}(n)>n/36$ .

Minimum Degree Conditions for the Proper Connection Number of Graphs

2017 ◽  
Vol 33 (4) ◽  
pp. 833-843 ◽  
Author(s):  
Christoph Brause ◽  
Trung Duy Doan ◽  
Ingo Schiermeyer
Keyword(s):  
Minimum Degree ◽  
Connection Number ◽  
Proper Connection Number ◽  
Degree Conditions

On the minimum degree and the proper connection number of graphs

2016 ◽  
Vol 55 ◽  
pp. 109-112 ◽  
Author(s):  
Christoph Brause ◽  
Trung Duy Doan ◽  
Ingo Schiermeyer
Keyword(s):  
Minimum Degree ◽  
Connection Number ◽  
Proper Connection Number

scholarly journals Minimum degree and size conditions for the proper connection number of graphs

2019 ◽  
Vol 352 ◽  
pp. 205-210
Author(s):  
Xiaxia Guan ◽  
Lina Xue ◽  
Eddie Cheng ◽  
Weihua Yang
Keyword(s):  
Minimum Degree ◽  
Connection Number ◽  
Proper Connection Number

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

scholarly journals On two conjectures about the proper connection number of graphs

2017 ◽  
Vol 340 (9) ◽  
pp. 2217-2222 ◽  
Author(s):  
Fei Huang ◽  
Xueliang Li ◽  
Zhongmei Qin ◽  
Colton Magnant ◽  
Kenta Ozeki
Keyword(s):  
Connection Number ◽  
Proper Connection Number

Proper connection of power graphs of finite groups

2020 ◽  
pp. 2150033
Author(s):  
Xuanlong Ma
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Undirected Graph ◽  
The Other ◽  
Connection Number ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Connection Number ◽  
A Finite Group

Let [Formula: see text] be a finite group. The power graph of [Formula: see text] is the undirected graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of [Formula: see text] is the subgraph of the power graph of [Formula: see text] obtained by deleting all edges [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are two distinct elements of [Formula: see text]. In this paper, we determine the proper connection number of the reduced power graph of [Formula: see text]. As an application, we also determine the proper connection number of the power graph of [Formula: see text].

scholarly journals A minimum degree condition forcing complete graph immersion

COMBINATORICA ◽  
2014 ◽  
Vol 34 (3) ◽  
pp. 279-298 ◽  
Author(s):  
Matt Devos ◽  
Zdeněk Dvořák ◽  
Jacob Fox ◽  
Jessica McDonald ◽  
Bojan Mohar ◽  
...  
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Degree Condition

scholarly journals A MINIMUM DEGREE CONDITION FOR FRACTIONAL ID-[a,b]-FACTOR-CRITICAL GRAPHS

2012 ◽  
Vol 86 (2) ◽  
pp. 177-183 ◽  
Author(s):  
SIZHONG ZHOU ◽  
ZHIREN SUN ◽  
HONGXIA LIU
Keyword(s):  
Minimum Degree ◽  
Independent Set ◽  
Indicator Function ◽  
Critical Graphs ◽  
Degree Condition

AbstractLet G be a graph of order n, and let a and b be two integers with 1≤a≤b. Let h:E(G)→[0,1] be a function. If a≤∑ e∋xh(e)≤b holds for any x∈V (G), then we call G[Fh] a fractional [a,b] -factor of G with indicator function h, where Fh ={e∈E(G):h(e)>0}. A graph G is fractional independent-set-deletable [a,b] -factor-critical (in short, fractional ID-[a,b] -factor-critical) if G−I has a fractional [a,b] -factor for every independent set I of G. In this paper, it is proved that if n≥((a+2b)(a+b−2)+1 )/b and δ(G)≥((a+b)n )/(a+2b ) , then G is fractional ID-[a,b] -factor-critical. This result is best possible in some sense, and it is an extension of Chang, Liu and Zhu’s previous result.

scholarly journals On the (di)graphs with (directed) proper connection number two

2017 ◽  
Vol 62 ◽  
pp. 237-242 ◽  
Author(s):  
Guillaume Ducoffe ◽  
Ruxandra Marinescu-Ghemeci ◽  
Alexandru Popa
Keyword(s):  
Connection Number ◽  
Proper Connection Number
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