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Coloring quasi-line graphs

2006
*
Journal of Graph Theory
*

The class of

doi:10.1002/jgt.20192
fatcat:hvnzardux5g5pomks7ntnbm3le
*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*[1] that we will use to prove our main results. Theorem 2.1. Let G be a connected,*quasi*-*line**graph*. ...##
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On-line coloring between two lines
[article]

2015
*
arXiv
*
pre-print

We present an on-

arXiv:1411.0402v2
fatcat:hwo3jcuqrffzzdt5acbiwdqfwm
*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. ...##
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Klein Group And Four Color Theorem
[article]

2010
*
arXiv
*
pre-print

In this work methods of construction of cubic

arXiv:1006.0276v2
fatcat:vqftg5tkd5exbdkwyrmicboc7m
*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. ...##
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Hadwiger's conjecture for quasi-line graphs

2008
*
Journal of Graph Theory
*

The class of

doi:10.1002/jgt.20321
fatcat:23weke3uxjanfasqtsutnbvxoi
*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 [10] . We extend their result to all*quasi*-*line**graphs*. ... We are finally ready to state the structure theorem for*quasi*-*line**graphs*[3] that we will use to prove our main result. Theorem 2.1. Let G be a connected,*quasi*-*line**graph*. ...##
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Claw-free circular-perfect graphs

2010
*
Journal of Graph Theory
*

Circular-perfect

doi:10.1002/jgt.20474
fatcat:55jb3rxrvrcn7pwwqe7s7ubg3m
*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*. ...##
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Claw-free circular-perfect graphs

2007
*
Electronic Notes in Discrete Mathematics
*

Circular-perfect

doi:10.1016/j.endm.2007.07.071
fatcat:tvthn7qjrrdmfmtrnkfa7vcjme
*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*. ...##
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Coloring claw-free graphs with Δ-1 colors
[article]

2012
*
arXiv
*
pre-print

We prove that every claw-free

arXiv:1206.1269v2
fatcat:7zxdtgb47rcz5po5dxvy6pgjui
*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*. ...##
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Some counterexamples associated with the three-color problem

1980
*
Journal of combinatorial theory. Series B (Print)
*

In this paper we construct some counterexamples of non-3-

doi:10.1016/0095-8956(80)90051-9
fatcat:xoxmwtbuknfathgmnunckzgvi4
*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 [4] , 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. ...##
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New bounds for the b-chromatic number of vertex deleted graphs
[article]

2019
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arXiv
*
pre-print

In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a

arXiv:1904.01600v1
fatcat:u3dpjwtkzzan7oz4bsnmaedvwe
*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*. ...##
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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*. ...##
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Quasi-planar graphs have a linear number of edges
[chapter]

1996
*
Lecture Notes in Computer Science
*

It is shown that the maximum number of edges of a

doi:10.1007/bfb0021784
fatcat:kb3gckettvae7e6ilome5ivcry
*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. ...##
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Edge Partitions of Complete Geometric Graphs (Part 2)
[article]

2021
*
arXiv
*
pre-print

Recently, the second and third author showed that complete geometric

arXiv:2112.08456v1
fatcat:3uchv25xufbzpcwspaz3mgw4ka
*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. ...##
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How to Draw a Tait-Colorable Graph
[chapter]

2011
*
Lecture Notes in Computer Science
*

Presented here are necessary and sufficient conditions for a cubic

doi:10.1007/978-3-642-18469-7_32
fatcat:4iwcxuprkrhy7abvc5lazbayc4
*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. ...##
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Polyhedral studies of vertex coloring problems: The asymmetric representatives formulation
[article]

2015
*
arXiv
*
pre-print

In this work we study the asymmetric representatives formulation and we show that the corresponding

arXiv:1509.02485v1
fatcat:kmjow2rs75cq7fqcjyc5ibmx64
*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. ...##
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Beyond Outerplanarity
[chapter]

2018
*
Lecture Notes in Computer Science
*

We study straight-

doi:10.1007/978-3-319-73915-1_42
fatcat:r523bjfdojbotfnm7pxuzkmvry
*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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