241 research outputs found
On graphs double-critical with respect to the colouring number
The colouring number col(G) of a graph G is the smallest integer k for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k-1 in the remaining graph is removed at any step. An edge e of a graph G is said to be &em;double-col-critical if the colouring number of G-V(e) is at most the colouring number of G minus 2. A connected graph G is said to be double-col-critical if each edge of G is double-col-critical. We characterise the double-col-critical graphs with colouring number at most 5. In addition, we prove that every 4-col-critical non-complete graph has at most half of its edges being double-col-critical, and that the extremal graphs are precisely the odd wheels on at least six vertices. We observe that for any integer k greater than 4 and any positive number ε, there is a k-col-critical graph with the ratio of double-col-critical edges between 1- ε and 1
Vertex colouring and forbidden subgraphs - a survey
There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions
Critical (,bull)-free graphs
Given two graphs and , a graph is -free if it contains
no induced subgraph isomorphic to or . Let and be the
path and the cycle on vertices, respectively. A bull is the graph obtained
from a triangle with two disjoint pendant edges. In this paper, we show that
there are finitely many 5-vertex-critical (,bull)-free graphs.Comment: 21 page
A characterization of b-chromatic and partial Grundy numbers by induced subgraphs
Gy{\'a}rf{\'a}s et al. and Zaker have proven that the Grundy number of a
graph satisfies if and only if contains an induced
subgraph called a -atom.The family of -atoms has bounded order and
contains a finite number of graphs.In this article, we introduce equivalents of
-atoms for b-coloring and partial Grundy coloring.This concept is used to
prove that determining if and (under
conditions for the b-coloring), for a graph , is in XP with parameter .We
illustrate the utility of the concept of -atoms by giving results on
b-critical vertices and edges, on b-perfect graphs and on graphs of girth at
least
On graphs with no induced or
In this paper, we are interested in some problems related to chromatic number
and clique number for the class of -free graphs, and prove the
following. If is a connected ()-free graph with
, then either is the complement of a bipartite graph or
has a clique cut-set. Moreover, there is a connected ()-free
imperfect graph with and has no clique cut-set. This
strengthens a result of Malyshev and Lobanova [Disc. Appl. Math. 219 (2017)
158--166]. If is a ()-free graph with ,
then . Moreover, the bound is tight when
. This result together with known results partially
answers a question of Ju and Huang [arXiv:2303.18003 [math.CO] 2023], and also
improves a result of Xu [Manuscript 2022].
While the "Chromatic Number Problem" is known to be -hard for the class
of -free graphs, our results together with some known results imply that
the "Chromatic Number Problem" can be solved in polynomial time for the class
of ()-free graphs which may be independent interest.Comment: This paper is dedicated to the memory of Professor Frederic Maffray
on his death anniversar
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