416 research outputs found
Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration
We continue research into a well-studied family of problems that ask whether
the vertices of a graph can be partitioned into sets and~, where is
an independent set and induces a graph from some specified graph class
. We let be the class of -degenerate graphs. This
problem is known to be polynomial-time solvable if (bipartite graphs) and
NP-complete if (near-bipartite graphs) even for graphs of maximum degree
. Yang and Yuan [DM, 2006] showed that the case is polynomial-time
solvable for graphs of maximum degree . This also follows from a result of
Catlin and Lai [DM, 1995]. We consider graphs of maximum degree on
vertices. We show how to find and in time for , and in
time for . Together, these results provide an algorithmic
version of a result of Catlin [JCTB, 1979] and also provide an algorithmic
version of a generalization of Brook's Theorem, which was proven in a more
general way by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007].
Moreover, the two results enable us to complete the complexity classification
of an open problem of Feghali et al. [JGT, 2016]: finding a path in the vertex
colouring reconfiguration graph between two given -colourings of a graph
of maximum degree
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
On Brooks' Theorem
In this note we give two proofs of Brooks' Theorem. The first is obtained by
modifying an earlier proof and the second by combining two earlier proofs. We
believe these proofs are easier to teach in Computer Science courses.Comment: 5 page
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