Precoloring extension with fixed color bound.
Kratochvíl, J. (1993)
Acta Mathematica Universitatis Comenianae. New Series
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Kratochvíl, J. (1993)
Acta Mathematica Universitatis Comenianae. New Series
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William F. Klostermeyer, Gary MacGillivray (2004)
Discussiones Mathematicae Graph Theory
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We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.
Xin Zhang, Yong Yu, Guizhen Liu (2011)
Open Mathematics
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A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that the (p, 1)-total labelling number of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ(G) ≥ 8p+4 or Δ(G) ≥ 6p+2 and g(G) ≥ 4. As a consequence, the well-known (p, 1)-total labelling conjecture has been confirmed for some 1-planar graphs.
Hujter, M., Tuza, Zs. (1993)
Acta Mathematica Universitatis Comenianae. New Series
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Zhu, Xuding (2001)
The Electronic Journal of Combinatorics [electronic only]
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Sebastian Urbański (1996)
Discussiones Mathematicae Graph Theory
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The paper gives an account of previous and recent attempts to determine the order of a smallest graph not containing K₅ and such that every 2-coloring of its edges results in a monochromatic triangle. A new 14-vertex K₄-free graph with the same Ramsey property in the vertex coloring case is found. This yields a new construction of one of the only two known 15-vertex (3,3)-Ramsey graphs not containing K₅.
Oleg V. Borodin, Anna O. Ivanova (2013)
Discussiones Mathematicae Graph Theory
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We prove that every planar graph with maximum degree ∆ is strong edge (2∆−1)-colorable if its girth is at least 40 [...] +1. The bound 2∆−1 is reached at any graph that has two adjacent vertices of degree ∆.
Jair Donadelli, Penny E. Haxell, Yoshiharu Kohayakawa (2005)
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
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Let be the graph obtained from a given graph by subdividing each edge times. Motivated by a problem raised by Igor Pak [Mixing time and long paths in graphs, in Proc. of the 13th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002) 321–328], we prove that, for any graph , there exist graphs with edges that are Ramsey with respect to .
Alishahi, Meysam, Taherkhani, Ali, Thomassen, Carsten (2011)
The Electronic Journal of Combinatorics [electronic only]
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Albertson, Michael O., Chappell, Glenn G., Kierstead, H.A., Kündgen, André, Ramamurthi, Radhika (2004)
The Electronic Journal of Combinatorics [electronic only]
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Hajo Broersma, Bert Marchal, Daniel Paulusma, A.N.M. Salman (2009)
Discussiones Mathematicae Graph Theory
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We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring for G and H is a proper vertex coloring V→ {1,2,...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers....
Xin Zhang, Guizhen Liu (2013)
Open Mathematics
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If a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity...
Gyárfás, András (1997)
The Electronic Journal of Combinatorics [electronic only]
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