New Bounds on the List-Chromatic Index of the Complete Graph and Other Simple Graphs

1997 ◽  
Vol 6 (3) ◽  
pp. 295-313 ◽  
Author(s):  
ROLAND HÄGGKVIST ◽  
JEANNETTE JANSSEN
Keyword(s):  
Complete Graph ◽  
Maximum Degree ◽  
Simple Graph ◽  
Asymptotic Result ◽  
Complete Graphs ◽  
Chromatic Index ◽  
Simple Graphs ◽  
Odd Order

In this paper we show that the list chromatic index of the complete graph Kn is at most n. This proves the list-chromatic conjecture for complete graphs of odd order. We also prove the asymptotic result that for a simple graph with maximum degree d the list chromatic index exceeds d by at most [Oscr ](d2/3√log d).

scholarly journals Vizing's 2-Factor Conjecture Involving Toughness and Maximum Degree Conditions

10.37236/7353 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Jinko Kanno ◽  
Songling Shan
Keyword(s):  
Maximum Degree ◽  
Hamiltonian Cycle ◽  
Simple Graph ◽  
Chromatic Index ◽  
New Techniques ◽  
Critical Graph ◽  
Degree Conditions ◽  
Delta G ◽  
Proper Subgraph

Let $G$ be a simple graph, and let $\Delta(G)$ and $\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\chi'(G)=\Delta(G)$ or $\chi'(G)=\Delta(G)+1$. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $\Delta$-critical graph, then  $G$ has a 2-factor. Let $G$ be an $n$-vertex $\Delta$-critical graph. It was proved that if $\Delta(G)\ge n/2$, then $G$ has a 2-factor; and that if $\Delta(G)\ge 2n/3+13$, then $G$  has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a $\Delta$-critical graph under "moderate" given toughness and  maximum degree conditions. In particular, we show that  if $G$ is an  $n$-vertex $\Delta$-critical graph with toughness at least 3/2 and with maximum degree at least $n/3$, then $G$ has a 2-factor. We also construct a family of graphs that have order $n$, maximum degree $n-1$, toughness at least $3/2$, but have no 2-factor. This implies that the $\Delta$-criticality in the result is needed. In addition, we develop new techniques in proving the existence of 2-factors in graphs.

scholarly journals Star multigraphs with three vertices of maximum degree

1986 ◽  
Vol 100 (2) ◽  
pp. 303-317 ◽  
Author(s):  
A. G. Chetwynd ◽  
A. J. W. Hilton
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Special Kind ◽  
Chromatic Index ◽  
Simple Graphs ◽  
Multiple Edges ◽  
Image Position ◽  
Edge Colouring

The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with each non-simple edge. An edge-colouring of a multigraph G is a map ø: E(G)→, where is a set of colours and E(G) is the set of edges of G, such that no two edges receiving the same colour have a vertex in common. The chromatic index, or edge-chromatic numberχ′(G) of G is the least value of || for which an edge-colouring of G exists. Generalizing a well-known theorem of Vizing [14], we showed in [6] that, for a star-multigraph G,where Δ(G) denotes the maximum degree (that is, the maximum number of edges incident with a vertex) of G. Star-multigraphs for which χ′(G) = Δ(G) are said to be Class 1, and otherwise they are Class 2.

scholarly journals Total chromatic number of graphs of high degree, II

1992 ◽  
Vol 53 (2) ◽  
pp. 219-228 ◽  
Author(s):  
H. P. Yap ◽  
K. H. Chew
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Independent Set ◽  
Simple Graph ◽  
Simple Graphs ◽  
Total Chromatic Number ◽  
Prove Theorem ◽  
High Degree

AbstractWe prove Theorem 1: suppose G is a simple graph of order n having Δ(G) = n − k where k ≥ 5 and n ≥ max (13, 3k −3). If G contains an independent set of k − 3 vertices, then the TCC (Total Colouring Conjecture) is true. Applying Theorem 1, we also prove that the TCC is true for any simple graph G of order n having Δ(G) = n −5. The latter result together with some earlier results confirm that the TCC is true for all simple graphs whose maximum degree is at most four and for all simple graphs of order n having maximum degree at least n − 5.

scholarly journals How to Find Overfull Subgraphs in Graphs with Large Maximum Degree, II

10.37236/1551 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Thomas Niessen
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Simple Graph ◽  
Chromatic Index ◽  
Large Maximum ◽  
Delta G

Let $G$ be a simple graph with $3\Delta (G) > |V|$. The Overfull Graph Conjecture states that the chromatic index of $G$ is equal to $\Delta (G)$, if $G$ does not contain an induced overfull subgraph $H$ with $\Delta (H) = \Delta (G)$, and otherwise it is equal to $\Delta (G) +1$. We present an algorithm that determines these subgraphs in $O(n^{5/3}m)$ time, in general, and in $O(n^3)$ time, if $G$ is regular. Moreover, it is shown that $G$ can have at most three of these subgraphs. If $2\Delta (G) \geq |V|$, then $G$ contains at most one of these subgraphs, and our former algorithm for this situation is improved to run in linear time.

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.

scholarly journals Strong chromatic index of products of graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Olivier Togni
Keyword(s):  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Index ◽  
Induced Matching ◽  
Colour Class ◽  
Strong Chromatic Index ◽  
Minimum Number ◽  
International Audience

Graphs and Algorithms International audience The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.

scholarly journals A Note on $K_{\Delta+1}^-$-Free Precolouring with $\Delta$ Colours

10.37236/266 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Tom Rackham
Keyword(s):  
Maximum Degree ◽  
Simple Graph ◽  
Pairwise Distance ◽  
Edge Connectivity ◽  
Extra Assumption

Let $G$ be a simple graph of maximum degree $\Delta \geq 3$, not containing $K_{\Delta + 1}$, and ${\cal L}$ a list assignment to $V(G)$ such that $|{\cal L}(v)| = \Delta$ for all $v \in V(G)$. Given a set $P \subset V(G)$ of pairwise distance at least $d$ then Albertson, Kostochka and West (2004) and Axenovich (2003) have shown that every ${\cal L}$-precolouring of $P$ extends to a ${\cal L}$-colouring of $G$ provided $d \geq 8$. Let $K_{\Delta + 1}^-$ denote the graph $K_{\Delta + 1}$ with one edge removed. In this paper, we consider the problem above and the effect on the pairwise distance required when we additionally forbid either $K_{\Delta + 1}^-$ or $K_{\Delta}$ as a subgraph of $G$. We have the corollary that an extra assumption of 3-edge-connectivity in the above result is sufficient to reduce this distance from $8$ to $4$. This bound is sharp with respect to both the connectivity and distance. In particular, this corrects the results of Voigt (2007, 2008) for which counterexamples are given.

scholarly journals Consecutive Colouring of Oriented Graphs

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Marta Borowiecka-Olszewska ◽  
Ewa Drgas-Burchardt ◽  
Nahid Yelene Javier-Nol ◽  
Rita Zuazua
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Oriented Graphs ◽  
Complete Problem ◽  
Np Complete

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

scholarly journals INTRINSICALLY LINKED GRAPHS WITH KNOTTED COMPONENTS

2012 ◽  
Vol 21 (07) ◽  
pp. 1250065 ◽  
Author(s):  
THOMAS FLEMING
Keyword(s):  
Complete Graph ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Linking Number ◽  
A Minor

We construct a graph G such that any embedding of G into R3 contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a 2-component link whose linking number is a nonzero multiple of n. Finally, we show that if a graph is a Cartesian product of the form G × K2, it is intrinsically linked if and only if G contains one of K5, K3,3 or K4,2 as a minor.

Finding a Longest Alternating Cycle in a 2-edge-coloured Complete Graph is in RP

1996 ◽  
Vol 5 (3) ◽  
pp. 297-306 ◽  
Author(s):  
Rachid Saad
Keyword(s):  
Complete Graph ◽  
Strong Evidence ◽  
General Setting ◽  
Maximum Length ◽  
Sufficient Condition ◽  
Complete Graphs ◽  
Random Polynomial ◽  
Exact Matching ◽  
Matching Problem ◽  
Bipartite Tournament

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

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