scholarly journals Bispindles in Strongly Connected Digraphs with Large Chromatic Number

10.37236/6922 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Nathann Cohen ◽  
Frédéric Havet ◽  
William Lochet ◽  
Raul Lopes
Keyword(s):  
Chromatic Number ◽  
The Other ◽  
Strongly Connected Digraph ◽  
Connected Digraph ◽  
Strongly Connected ◽  
Positive Integers ◽  
Large Chromatic Number

A $(k_1+k_2)$-bispindle is the union of $k_1$ $(x,y)$-dipaths and $k_2$ $(y,x)$-dipaths, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. showed that for every $(1,1)$- bispindle $B$, there exists an integer $k$ such that every strongly connected digraph with chromatic number greater than $k$ contains a subdivision of $B$. We investigate generalizations of this result by first showing constructions of strongly connected digraphs with large chromatic number without any $(3,0)$-bispindle or $(2,2)$-bispindle. We then consider $(2,1)$-bispindles. Let $B(k_1,k_2;k_3)$ denote the $(2,1)$-bispindle formed by three internally disjoint dipaths between two vertices $x,y$, two $(x,y)$-dipaths, one of length $k_1$ and the other of length $k_2$, and one $(y,x)$-dipath of length $k_3$. We conjecture that for any positive integers $k_1, k_2,k_3$, there is an integer $g(k_1,k_2,k_3)$ such that every strongly connected digraph with chromatic number greater than $g(k_1,k_2,k_3)$ contains a subdivision of $B(k_1,k_2;k_3)$. As evidence, we prove this conjecture for $k_2=1$ (and $k_1, k_3$ arbitrary).

Restricted connectivity of total digraph

2016 ◽  
Vol 08 (02) ◽  
pp. 1650022
Author(s):  
Zhenhua Liu ◽  
Zhao Zhang
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Vertex Connectivity ◽  
Minimum Cardinality ◽  
Strongly Connected Digraph ◽  
Connected Digraph ◽  
Strongly Connected

For a strongly connected digraph [Formula: see text], an arc-cut [Formula: see text] is a [Formula: see text]-restricted arc-cut of [Formula: see text] if [Formula: see text] has at least two strong components of order at least [Formula: see text]. The [Formula: see text]-restricted arc-connectivity [Formula: see text] is the minimum cardinality of all [Formula: see text]-restricted arc-cuts. The [Formula: see text]-restricted vertex-connectivity [Formula: see text] can be defined similarly. In this paper, we provide upper and lower bounds for [Formula: see text] and [Formula: see text] for the total digraph [Formula: see text] of [Formula: see text].

scholarly journals Short Cycles in Digraphs with Local Average Outdegree at Least Two

10.37236/1719 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Jian Shen
Keyword(s):  
Strongly Connected Digraph ◽  
Connected Digraph ◽  
Strongly Connected ◽  
Weaker Conditions ◽  
Local Average

Suppose $G$ is a strongly connected digraph with order $n$ girth $g$ and diameter $d$. We prove that $d +g \le n$ if $G$ contains no arcs $(u,v)$ with $\deg^+(u)=1$ and $\deg^+(v) \le 2$.Caccetta and H${\rm\ddot{a}}$ggkvist showed in 1978 that any digraph of order $n$ with minimum outdegree $2$ contains a cycle of length at most $\lceil n/2 \rceil$. Applying the above-mentioned result, we improve their result by replacing the minimum outdegree condition by some weaker conditions involving the local average outdegree. In particular, we prove that, for any digraph $G$ of order $n$, if either(1) $G$ has minimum outdegree $1$ and $\deg^+(u) +\deg^+(v) \ge 4$ for all arcs $(u,v)$, or(2) $\deg^+(u) +\deg^+(v) \ge 3$ for all pairs of distinct vertices $u,v$,then $G$ contains a cycle of length at most $\lceil n/2 \rceil$.

A Note on p-Cyclic Matrices and Digraphs

1967 ◽  
Vol 10 (4) ◽  
pp. 497-501
Author(s):  
B. R. Heap ◽  
M. S Lynn
Keyword(s):  
Adjacency Matrix ◽  
Greatest Common Divisor ◽  
Strongly Connected Digraph ◽  
Connected Digraph ◽  
Strongly Connected ◽  
Directed Cycles

We use the terminology of [1]. Let D be a strongly connected digraph on n points and containing m lines, and let A = A(D) be the corresponding adjacency matrix, so that A is an n x n 0-1 matrix containing m unit elements. We recall that A and D are said to be p-cyclic if p is the greatest common divisor of the lengths of all directed cycles of D. Clearly, the larger the value of p, the smaller the value of m must be; in this note we make the latter and related statements precise.

scholarly journals Bispindle in strongly connected digraphs with large chromatic number

2017 ◽  
Vol 62 ◽  
pp. 69-74
Author(s):  
Nathann Cohen ◽  
Frédéric Havet ◽  
William Lochet ◽  
Raul Lopes
Keyword(s):  
Chromatic Number ◽  
Strongly Connected ◽  
Large Chromatic Number

scholarly journals A Min-Max theorem about the Road Coloring Conjecture

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Rajneesh Hegde ◽  
Kamal Jain
Keyword(s):  
Prime Number ◽  
Prime Number Theorem ◽  
Necessary Condition ◽  
Blue Edge ◽  
The Road ◽  
Strongly Connected Digraph ◽  
Connected Digraph ◽  
Strongly Connected ◽  
International Audience

International audience The Road Coloring Conjecture is an old and classical conjecture e posed in Adler and Weiss (1970); Adler et al. (1977). Let $G$ be a strongly connected digraph with uniform out-degree $2$. The Road Coloring Conjecture states that, under a natural (necessary) condition that $G$ is "aperiodic'', the edges of $G$ can be colored red and blue such that "universal driving directions'' can be given for each vertex. More precisely, each vertex has one red and one blue edge leaving it, and for any vertex $v$ there exists a sequence $s_v$ of reds and blues such that following the sequence from $\textit{any}$ starting vertex in $G$ ends precisely at the vertex $v$. We first generalize the conjecture to a min-max conjecture for all strongly connected digraphs. We then generalize the notion of coloring itself. Instead of assigning exactly one color to each edge we allow multiple colors to each edge. Under this relaxed notion of coloring we prove our generalized Min-Max theorem. Using the Prime Number Theorem (PNT) we further show that the number of colors needed for each edge is bounded above by $O(\log n / \log \log n)$, where $n$ is the number of vertices in the digraph.

On the Diameter of a p-Cyclic Strongly Connected Digraph

1968 ◽  
Vol 20 ◽  
pp. 749-755
Author(s):  
M. Stuart Lynn
Keyword(s):  
Upper Bounds ◽  
Strongly Connected Digraph ◽  
Connected Digraph ◽  
Strongly Connected ◽  
Image Position

In this paper we follow the notation of (2). In (5), Luce showed, in other terminology, that if d is the diameter of a strongly connected digraph, D, on n vertices with m edges, then1.1this inequality being sharp; from (1.1) one may immediately derive sharp upper bounds for d in terms of m and n, this being a generalization of the obvious and well-known inequality1.2

scholarly journals Upper bound for the energy of strongly connected digraphs

2011 ◽  
Vol 5 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Singaraj Ayyaswamy ◽  
Selvaraj Balachandran ◽  
Ivan Gutman
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Strongly Regular Graph ◽  
Simple Graphs ◽  
Strongly Connected Digraph ◽  
Complex Eigenvalues ◽  
Connected Digraph ◽  
Strongly Connected ◽  
Strongly Regular ◽  
Appl Math

The energy of a digraph D is defined as E(D) = ?n,i=1 ?Re(zi)?, where z1, z2, ..., zn are the (possibly complex) eigenvalues of D . We show that if D is a strongly connected digraph on n vertices, a arcs, and c2 closed walks of length two, such that Re(z1) ? (a + c2)=(2n) ? 1 , then E(D) ? n(1 + ?n)=2. Equality holds if and only if D is a directed strongly regular graph with parameters (n, n+?n/2, 3n+2?n/8, n+2?n/8, n+2?n/8). This bound extends to digraphs an earlier result [J. H. Koolen, V. Moulton:, Maximal energy graphs. Adv. Appl. Math., 26 (2001), 47-52], obtained for simple graphs.

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