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### Coloring quasi-line graphs

Maria Chudnovsky, Alexandra Ovetsky
2006 Journal of Graph Theory
The class of quasi-line graphs is a proper superset of the class of line graphs.  ...  A graph G is a quasi-line graph if for every vertex v, the set of neighbors of v can be expressed as the union of two cliques.  ...  We are finally ready to state the structure theorem for quasi-line graphs  that we will use to prove our main results. Theorem 2.1. Let G be a connected, quasi-line graph.  ...

### On-line coloring between two lines [article]

Stefan Felsner, Piotr Micek, Torsten Ueckerdt
2015 arXiv   pre-print
We present an on-line algorithm coloring graphs given by convex sets between two lines that uses O(ω^3) colors on graphs with maximum clique size ω.  ...  In contrast intersection graphs of segments attached to a single line may force any on-line coloring algorithm to use an arbitrary number of colors even when ω=2.  ...  We describe an on-line coloring algorithm using at most ω+1 2 · 24ω colors on quasi-convex sets spanned between two parallel lines with clique number at most ω.  ...

### Klein Group And Four Color Theorem [article]

Sergey Kurapov
2010 arXiv   pre-print
In this work methods of construction of cubic graphs are analyzed and a theorem of existence of a colored disc traversing each pair of linked edges belonging to an elementary cycle of a planar cubic graph  ...  as dot lines.  ...  This makes the cubic graph H colored properly and induces three colored Hamiltonian quasi-cycles. Property 3.  ...

### Hadwiger's conjecture for quasi-line graphs

2008 Journal of Graph Theory
The class of quasi-line graphs is a proper superset of the class of line graphs. Hadwiger's conjecture states that if a graph G is not t-colorable then it contains K t+1 as a minor.  ...  This conjecture has been proved for line graphs by Reed and Seymour  . We extend their result to all quasi-line graphs.  ...  We are finally ready to state the structure theorem for quasi-line graphs  that we will use to prove our main result. Theorem 2.1. Let G be a connected, quasi-line graph.  ...

### Claw-free circular-perfect graphs

Arnaud Pêcher, Xuding Zhu
2010 Journal of Graph Theory
Circular-perfect graphs is a superclass of perfect graphs defined by means of this more general coloring concept. This paper studies claw-free circular-perfect graphs.  ...  A consequence of the strong perfect graph theorem is that minimal circular-imperfect graphs G have min{α(G), ω(G)} = 2.  ...  A graph G for which the neighbourhood of each vertex can be covered by two cliques is called a quasi-line graph.  ...

### Claw-free circular-perfect graphs

Arnaud Pêcher, Xuding Zhu
2007 Electronic Notes in Discrete Mathematics
Circular-perfect graphs is a superclass of perfect graphs defined by means of this more general coloring concept. This paper studies claw-free circular-perfect graphs.  ...  A consequence of the strong perfect graph theorem is that minimal circular-imperfect graphs G have min{α(G), ω(G)} = 2.  ...  A graph G for which the neighbourhood of each vertex can be covered by two cliques is called a quasi-line graph.  ...

### Coloring claw-free graphs with Δ-1 colors [article]

Daniel W. Cranston, Landon Rabern
2012 arXiv   pre-print
We prove that every claw-free graph G that doesn't contain a clique on Δ(G) ≥ 9 vertices can be Δ(G) - 1 colored.  ...  " smaller quasi-line graphs.  ...  Quasi-line graphs are a proper subset of clawfree graphs and a proper superset of line graphs.  ...

### Some counterexamples associated with the three-color problem

V.A Aksionov, L.S Mel'nikov
1980 Journal of combinatorial theory. Series B (Print)
In this paper we construct some counterexamples of non-3-colorable planar graphs, using the notion of "quasi-edges." The minimality of some quasi-edges is proved.  ...  Our graph theoretic terminology is that of Harary  , except that we use vertices and edges instead of points and lines, respectively.  ...  On the other hand, if we have a quasi-edge we may obtain a non-3-colorable graph by identification of end-vertices of the quasi-edge; thus we obtain a quasi-loop.  ...

### New bounds for the b-chromatic number of vertex deleted graphs [article]

Renata Del-Vecchio, Mekkia Kouider
2019 arXiv   pre-print
In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a graph, particularly regarding two important classes, quasi-line and chordal graphs.  ...  The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors.  ...  , quasi-line and chordal graphs.  ...

### Page 635 of Mathematical Reviews Vol. , Issue 94b [page]

1994 Mathematical Reviews
This paper shows that the strong perfect graph conjecture is valid for 3-line graphs and 3-total graphs.  ...  The quasi-brittle graphs turn out to be a natural generalization of the well-known class of brittle graphs.  ...

### Quasi-planar graphs have a linear number of edges [chapter]

Pankaj K. Agarwal, Boris Aronov, János Pach, Richard Pollack, Micha Sharir
1996 Lecture Notes in Computer Science
It is shown that the maximum number of edges of a quasi-planar graph with n vertices is O(n).  ...  A graph is called quasi-planar if it can be drawn in the plane so that no three of its edges are pairwise crossing.  ...  The third line is Euler's relation. Let G(V, E) be a quasi-planar graph drawn in the plane.  ...

### Edge Partitions of Complete Geometric Graphs (Part 2) [article]

Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, Birgit Vogtenhuber
2021 arXiv   pre-print
Recently, the second and third author showed that complete geometric graphs on 2n vertices in general cannot be partitioned into n plane spanning trees.  ...  Building up on this work, in this paper, we initiate the study of partitioning into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs  ...  Then, at least m k−1 colors are required and at most m k−1 + n−2m k−1 colors are needed to partition the complete geometric graph K(P ) into k-quasi-planar subgraphs.  ...

### How to Draw a Tait-Colorable Graph [chapter]

David A. Richter
2011 Lecture Notes in Computer Science
Presented here are necessary and sufficient conditions for a cubic graph equipped with a Tait-coloring to have a drawing in the real projective plane where every edge is represented by a line segment,  ...  all of the lines supporting the edges sharing a common color are concurrent, and all of the supporting lines are distinct.  ...  Quasi-Faithful Drawings A faithful projective drawing of a Tait-colored graph is the nicest possible because all supporting lines are distinct.  ...

### Polyhedral studies of vertex coloring problems: The asymmetric representatives formulation [article]

Victor Campos, Ricardo C. Corrêa, Diego Delle Donne, Javier Marenco, Annegret Wagler
2015 arXiv   pre-print
In this work we study the asymmetric representatives formulation and we show that the corresponding coloring polytope, for a given graph G, can be interpreted as the stable set polytope of another graph  ...  Despite the fact that some vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view.  ...  Semi-line graphs are either line graphs or quasi-line graphs without a representation as a FCIG.  ...

### Beyond Outerplanarity [chapter]

Steven Chaplick, Myroslav Kryven, Giuseppe Liotta, Andre Löffler, Alexander Wolff
2018 Lecture Notes in Computer Science
We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings.  ...  We show that the outer k-planar graphs are ( √ 4k + 1 + 1)-degenerate, and consequently that every outer k-planar graph can be ( √ 4k + 1 +2)colored, and this bound is tight.  ...  Each outer k-planar graph is √ 4k + 1 + 2 colorable. This is tight. Quasi-polynomial time recognition via balanced separators.  ...
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