scholarly journals A Superlocal Version of Reed's Conjecture

10.37236/2666 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Katherine Edwards ◽  
Andrew D. King
Keyword(s):  
Chromatic Number ◽  
Stable Set ◽  
Line Graphs ◽  
Stability Number ◽  
Single Vertex ◽  
Fractional Relaxation

Reed's well-known $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisfies $\chi \leq \lceil \frac 12(\Delta+1+\omega)\rceil$. The second author formulated a local strengthening of this conjecture that considers a bound supplied by the neighbourhood of a single vertex. Following the idea that the chromatic number cannot be greatly affected by any particular stable set of vertices, we propose a further strengthening that considers a bound supplied by the neighbourhoods of two adjacent vertices. We provide some fundamental evidence in support, namely that the stronger bound holds in the fractional relaxation and holds for both quasi-line graphs and graphs with stability number two. We also conjecture that in the fractional version, we can push the locality even further.

r-Dynamic Chromatic Number of Some Line Graphs

2018 ◽  
Vol 49 (4) ◽  
pp. 591-600 ◽  
Author(s):  
Hanna Furmańczyk ◽  
J. Vernold Vivin ◽  
N. Mohanapriya
Keyword(s):  
Chromatic Number ◽  
Line Graphs

scholarly journals A Strengthening of Brooks' Theorem for Line Graphs

10.37236/632 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Clique Number ◽  
Line Graphs ◽  
Delta G

We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.

scholarly journals The stable set polytope of quasi-line graphs

COMBINATORICA ◽  
2008 ◽  
Vol 28 (1) ◽  
pp. 45-67 ◽  
Author(s):  
Friedrich Eisenbrand ◽  
Gianpaolo Oriolo ◽  
Gautier Stauffer ◽  
Paolo Ventura
Keyword(s):  
Stable Set ◽  
Line Graphs ◽  
Stable Set Polytope

scholarly journals Stability number and chromatic number of tolerance graphs

1992 ◽  
Vol 36 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Giri Narasimhan ◽  
Rachel Manber
Keyword(s):  
Chromatic Number ◽  
Stability Number ◽  
Tolerance Graphs

scholarly journals An upper bound for the chromatic number of line graphs

2007 ◽  
Vol 28 (8) ◽  
pp. 2182-2187 ◽  
Author(s):  
A.D. King ◽  
B.A. Reed ◽  
A. Vetta
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Line Graphs

scholarly journals Squared Chromatic Number Without Claws or Large Cliques

2019 ◽  
Vol 62 (1) ◽  
pp. 23-35
Author(s):  
Wouter Cames van Batenburg ◽  
Ross J. Kang
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Clique Number ◽  
Sharp Bound ◽  
Line Graphs ◽  
Free Graph ◽  
Simple Graphs ◽  
Strengthened Form

AbstractLet $G$ be a claw-free graph on $n$ vertices with clique number $\unicode[STIX]{x1D714}$, and consider the chromatic number $\unicode[STIX]{x1D712}(G^{2})$ of the square $G^{2}$ of $G$. Writing $\unicode[STIX]{x1D712}_{s}^{\prime }(d)$ for the supremum of $\unicode[STIX]{x1D712}(L^{2})$ over the line graphs $L$ of simple graphs of maximum degree at most $d$, we prove that $\unicode[STIX]{x1D712}(G^{2})\leqslant \unicode[STIX]{x1D712}_{s}^{\prime }(\unicode[STIX]{x1D714})$ for $\unicode[STIX]{x1D714}\in \{3,4\}$. For $\unicode[STIX]{x1D714}=3$, this implies the sharp bound $\unicode[STIX]{x1D712}(G^{2})\leqslant 10$. For $\unicode[STIX]{x1D714}=4$, this implies $\unicode[STIX]{x1D712}(G^{2})\leqslant 22$, which is within 2 of the conjectured best bound. This work is motivated by a strengthened form of a conjecture of Erdős and Nešetřil.

scholarly journals On the maximum orders of an induced forest, an induced tree, and a stable set

2014 ◽  
Vol 24 (2) ◽  
pp. 199-215
Author(s):  
Alain Hertz ◽  
Odile Marcotte ◽  
David Schindl
Keyword(s):  
Connected Graph ◽  
Stable Set ◽  
Stability Number ◽  
Maximum Order ◽  
The Stability ◽  
Special Case

Let G be a connected graph, n the order of G, and f (resp. t) the maximum order of an induced forest (resp. tree) in G. We show that f - t is at most n - ?2?n-1?. In the special case where n is of the form a2 + 1 for some even integer a ? 4, f - t is at most n - ?2?n-1?-1. We also prove that these bounds are tight. In addition, letting ? denote the stability number of G, we show that ? - t is at most n + 1- ?2?2n? this bound is also tight.

The Strong Perfect Graph Conjecture for Planar Graphs

1973 ◽  
Vol 25 (1) ◽  
pp. 103-114 ◽  
Author(s):  
Alan Tucker
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Independent Set ◽  
Independent Sets ◽  
Clique Number ◽  
Perfect Graph ◽  
Stability Number ◽  
Partition Number ◽  
Minimum Number ◽  
The Stability

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).

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