scholarly journals Neighbour$\,$–$\,$Distinguishing Edge Colourings of Random Regular Graphs

10.37236/1103 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Andrzej Ruciński
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Edge Colourings ◽  
Almost All ◽  
Edge Colouring

A proper edge colouring of a graph is neighbour-distinguishing if for all pairs of adjacent vertices $v$, $w$ the set of colours appearing on the edges incident with $v$ is not equal to the set of colours appearing on the edges incident with $w$. Let ${\rm ndi}(G)$ be the least number of colours required for a proper neighbour-distinguishing edge colouring of $G$. We prove that for $d\geq 4$, a random $d$-regular graph $G$ on $n$ vertices asymptotically almost surely satisfies ${\rm ndi}(G)\leq \lceil 3d/2\rceil$. This verifies a conjecture of Zhang, Liu and Wang for almost all 4-regular graphs.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

scholarly journals Subgraphs of Randomk-Edge-Colouredk-Regular Graphs

2009 ◽  
Vol 18 (4) ◽  
pp. 533-549 ◽  
Author(s):  
PAULETTE LIEBY ◽  
BRENDAN D. McKAY ◽  
JEANETTE C. McLEOD ◽  
IAN M. WANLESS
Keyword(s):  
Regular Graph ◽  
Asymptotic Estimate ◽  
Forthcoming Paper ◽  
Perfect Matchings ◽  
Regular Graphs ◽  
Random Set ◽  
Expected Number ◽  
Edge Disjoint ◽  
Edge Colouring

LetG=G(n) be a randomly chosenk-edge-colouredk-regular graph with 2nvertices, wherek=k(n). Such a graph can be obtained from a random set ofkedge-disjoint perfect matchings ofK2n. Leth=h(n) be a graph withm=m(n) edges such thatm2+mk=o(n). Using a switching argument, we find an asymptotic estimate of the expected number of subgraphs ofGisomorphic toh. Isomorphisms may or may not respect the edge colouring, and other generalizations are also presented. Special attention is paid to matchings and cycles.The results in this paper are essential to a forthcoming paper of McLeod in which an asymptotic estimate for the number ofk-edge-colouredk-regular graphs fork=o(n5/6) is found.

scholarly journals Total Graph Interpretation of the Numbers of the Fibonacci Type

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Urszula Bednarz ◽  
Iwona Włoch ◽  
Małgorzata Wołowiec-Musiał
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Total Graph ◽  
Graph Interpretation ◽  
Edge Colourings ◽  
Almost All ◽  
Edge Colouring

We give a total graph interpretation of the numbers of the Fibonacci type. This graph interpretation relates to an edge colouring by monochromatic paths in graphs. We will show that it works for almost all numbers of the Fibonacci type. Moreover, we give the lower bound and the upper bound for the number of all(A1,2A1)-edge colourings in trees.

The maximum number of vertices of primitive regular graphs of orders 2, 3, 4 with exponent 2

2021 ◽  
pp. 97-104
Author(s):  
M. B. Abrosimov ◽  
◽  
S. V. Kostin ◽  
I. V. Los ◽  
◽  
...  
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Similar Question

In 2015, the results were obtained for the maximum number of vertices nk in regular graphs of a given order k with a diameter 2: n2 = 5, n3 = 10, n4 = 15. In this paper, we investigate a similar question about the largest number of vertices npk in a primitive regular graph of order k with exponent 2. All primitive regular graphs with exponent 2, except for the complete one, also have diameter d = 2. The following values were obtained for primitive regular graphs with exponent 2: np2 = 3, np3 = 4, np4 = 11.

Minimal Regular Graphs of Girths Eight and Twelve

1966 ◽  
Vol 18 ◽  
pp. 1091-1094 ◽  
Author(s):  
Clark T. Benson
Keyword(s):  
Group Theory ◽  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

scholarly journals On the Number of Spanning Trees in Random Regular Graphs

10.37236/3752 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Matthew Kwan ◽  
David Wind
Keyword(s):  
Asymptotic Distribution ◽  
Regular Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Expected Number ◽  
Fixed Integer ◽  
Numerical Evidence ◽  
Number Of Spanning Trees

Let $d\geq 3$ be a fixed integer.   We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

scholarly journals On Regular Factors in Regular Graphs with Small Radius

10.37236/1760 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Arne Hoffmann ◽  
Lutz Volkmann
Keyword(s):  
Regular Graph ◽  
Small Radius ◽  
Regular Graphs ◽  
High Eccentricity ◽  
K Factor

In this note we examine the connection between vertices of high eccentricity and the existence of $k$-factors in regular graphs. This leads to new results in the case that the radius of the graph is small ($\leq 3$), namely that a $d$-regular graph $G$ has all $k$-factors, for $k|V(G)|$ even and $k\le d$, if it has at most $2d+2$ vertices of eccentricity $>3$. In particular, each regular graph $G$ of diameter $\leq3$ has every $k$-factor, for $k|V(G)|$ even and $k\le d$.

scholarly journals Smallest regular graphs without near 1-factors

1986 ◽  
Vol 41 (2) ◽  
pp. 193-210 ◽  
Author(s):  
Gary Chartrand ◽  
Sergio Ruiz ◽  
Curtiss E. Wall
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Subgraph Isomorphic ◽  
Even Order

AbstractA near 1-factor of a graph of order 2n ≧ 4 is a subgraph isomorphic to (n − 2) K2 ∪ P3 ∪ K1. Wallis determined, for each r ≥ 3, the order of a smallest r-regular graph of even order without a 1-factor; while for each r ≧ 3, Chartrand, Goldsmith and Schuster determined the order of a smallest r-regular, (r − 2)-edge-connected graph of even order without a 1-factor. These results are extended to graphs without near 1-factors. It is known that every connected, cubic graph with less than six bridges has a near 1-factor. The order of a smallest connected, cubic graph with exactly six bridges and no near 1-factor is determined.

On the Non-Existence of a Type of Regular Graphs of Girth 5

1967 ◽  
Vol 19 ◽  
pp. 644-648 ◽  
Author(s):  
William G. Brown
Keyword(s):  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Minimal Graph ◽  
Image Position

ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that1.1Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.

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