scholarly journals A Stronger Bound for the Strong Chromatic Index

2017 ◽  
Vol 27 (1) ◽  
pp. 21-43 ◽  
Author(s):  
HENNING BRUHN ◽  
FELIX JOOS
Keyword(s):  
Maximum Degree ◽  
Type Inequality ◽  
Chromatic Index ◽  
Strong Chromatic Index ◽  
Large Maximum

We prove χ′s(G) ≤ 1.93 Δ(G)2 for graphs of sufficiently large maximum degree where χ′s(G) is the strong chromatic index of G. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where we are allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.

scholarly journals The Distance-t Chromatic Index of Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 90-101 ◽  
Author(s):  
TOMÁŠ KAISER ◽  
ROSS J. KANG
Keyword(s):  
Maximum Degree ◽  
Upper Bounds ◽  
The Other ◽  
Graph Colouring ◽  
Chromatic Index ◽  
Multiplicative Factor ◽  
Strong Chromatic Index ◽  
Absolute Positive Constant ◽  
Large Maximum ◽  
Fixed Positive Integer

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-ε)Δt for graphs of maximum degree at most Δ, where ε is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

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.

scholarly journals On the Strong Chromatic Index of Sparse Graphs

10.37236/5390 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Philip DeOrsey ◽  
Michael Ferrara ◽  
Nathan Graber ◽  
Stephen G. Hartke ◽  
Luke L. Nelsen ◽  
...  
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Edge Coloring ◽  
Combinatorial Nullstellensatz ◽  
Chromatic Index ◽  
Sparse Graphs ◽  
Strong Chromatic Index ◽  
General Bound

The strong chromatic index of a graph $G$, denoted $\chi'_s(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $\chi'_{s,\ell}(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors.We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with ${\rm girth}(G) \geq 41$ then $\chi'_{s,\ell}(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and ${\rm girth}(G) \geq 30$, then $\chi_s'(G) \leq 5$, improving a bound from the same paper.Finally, if $G$ is a planar graph with maximum degree at most four and ${\rm girth}(G) \geq 28$, then $\chi'_s(G)N \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its applications to the strong edge coloring, Applied Mathematics and Computation, 325 (2018), 246-251] in this case.

scholarly journals Strong Chromatic Index of Graphs With Maximum Degree Four

10.37236/7016 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Mingfang Huang ◽  
Michael Santana ◽  
Gexin Yu
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Recent Progress ◽  
Edge Coloring ◽  
Chromatic Index ◽  
Probabilistic Argument ◽  
New Approaches ◽  
Strong Chromatic Index ◽  
Color Class ◽  
Minimum Number

A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In 1985, Erdős and Nešetřil conjectured that every graph with maximum degree $\Delta$ has a strong edge-coloring using at most $\frac{5}{4}\Delta^2$ colors if $\Delta$ is even, and at most $\frac{5}{4}\Delta^2 - \frac{1}{2}\Delta + \frac{1}{4}$ if $\Delta$ is odd. Despite recent progress for large $\Delta$ by using an iterative probabilistic argument, the only nontrivial case of the conjecture that has been verified is when $\Delta = 3$, leaving the need for new approaches to verify the conjecture for any $\Delta\ge 4$. In this paper, we apply some ideas used in previous results to an upper bound of 21 for graphs with maximum degree 4, which improves a previous bound due to Cranston in 2006 and moves closer to the conjectured upper bound of 20.

scholarly journals Strong Cliques in Claw-Free Graphs

2021 ◽  
Author(s):  
Michał Dębski ◽  
Małgorzata Śleszyńska-Nowak
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Maximum Size ◽  
Chromatic Index ◽  
Free Graph ◽  
Famous Conjecture ◽  
Strong Chromatic Index

AbstractFor a graph G, $$L(G)^2$$ L ( G ) 2 is the square of the line graph of G – that is, vertices of $$L(G)^2$$ L ( G ) 2 are edges of G and two edges $$e,f\in E(G)$$ e , f ∈ E ( G ) are adjacent in $$L(G)^2$$ L ( G ) 2 if at least one vertex of e is adjacent to a vertex of f and $$e\ne f$$ e ≠ f . The strong chromatic index of G, denoted by $$s'(G)$$ s ′ ( G ) , is the chromatic number of $$L(G)^2$$ L ( G ) 2 . A strong clique in G is a clique in $$L(G)^2$$ L ( G ) 2 . Finding a bound for the maximum size of a strong clique in a graph with given maximum degree is a problem connected to a famous conjecture of Erdős and Nešetřil concerning strong chromatic index of graphs. In this note we prove that a size of a strong clique in a claw-free graph with maximum degree $$\varDelta $$ Δ is at most $$\varDelta ^2 + \frac{1}{2}\varDelta $$ Δ 2 + 1 2 Δ . This result improves the only known result $$1.125\varDelta ^2+\varDelta $$ 1.125 Δ 2 + Δ , which is a bound for the strong chromatic index of claw-free graphs.

scholarly journals The strong chromatic index of (3,Δ)-bipartite graphs

2017 ◽  
Vol 340 (5) ◽  
pp. 1143-1149 ◽  
Author(s):  
Mingfang Huang ◽  
Gexin Yu ◽  
Xiangqian Zhou
Keyword(s):  
Bipartite Graphs ◽  
Chromatic Index ◽  
Strong Chromatic Index

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.

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