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

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 $\unicode[STIX]{x1D6FF}(n)>n/36$ .

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

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

scholarly journals Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

10.37236/3610 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Louis DeBiasio ◽  
Theodore Molla
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Hamiltonian Cycles ◽  
Alternate Direction ◽  
Degree Conditions

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is 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 Maximum and Minimum Degree Conditions for Embedding Trees

2020 ◽  
Vol 34 (4) ◽  
pp. 2108-2123
Author(s):  
Guido Besomi ◽  
Matías Pavez-Signé ◽  
Maya Stein
Keyword(s):  
Minimum Degree ◽  
Degree Conditions ◽  
Maximum And Minimum Degree

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

Proper connection number 2, connectivity, and forbidden subgraphs

2016 ◽  
Vol 55 ◽  
pp. 105-108 ◽  
Author(s):  
Christoph Brause ◽  
Trung Duy Doan ◽  
Ingo Schiermeyer
Keyword(s):  
Forbidden Subgraphs ◽  
Connection Number ◽  
Proper Connection Number
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