5 research outputs found
An inertial lower bound for the chromatic number of a graph
Let ) and denote the chromatic and fractional chromatic
numbers of a graph , and let denote the inertia of .
We prove that:
1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G) \mbox{
and conjecture that } 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right)
\le \chi_f(G)
We investigate extremal graphs for these bounds and demonstrate that this
inertial bound is not a lower bound for the vector chromatic number. We
conclude with a discussion of asymmetry between and , including some
Nordhaus-Gaddum bounds for inertia
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Distances and Domination in Graphs
This book presents a compendium of the 10 articles published in the recent Special Issue âDistance and Domination in Graphsâ. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networksâ curvatures is also present
NordhausâGaddum inequalities for the fractional and circular chromatic numbers
AbstractFor a graph G on n vertices with chromatic number Ï(G), the NordhausâGaddum inequalities state that â2nââ€Ï(G)+Ï(GÂŻ)â€n+1, and nâ€Ï(G)â
Ï(GÂŻ)â€â(n+12)2â. Much analysis has been done to derive similar inequalities for other graph parameters, all of which are integer-valued. We determine here the optimal NordhausâGaddum inequalities for the circular chromatic number and the fractional chromatic number, the first examples of NordhausâGaddum inequalities where the graph parameters are rational-valued