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1,808
Colorings and Girth of Oriented Planar Graphs
 Discrete Math
, 1995
"... Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number. We investigate here homomorphisms from orientations of undirected planar graphs to graphs (not necessarily planar) containing as few digons as possible. We relate the existence of such homomorphisms to ..."
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Cited by 12 (7 self)
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to girth and it appears that these questions remain interesting even under large girth assumption in the range where the chromatic number is an easy invariant. In particular we prove that every orientation of any large girth planar graph is 5colorable and classify those digraphs on 3, 4 and 5 vertices
An oriented coloring of planar graphs with girth at least five
 DISCRETE MATH
, 2008
"... An oriented coloring of planar graphs with girth at least five ..."
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Cited by 2 (2 self)
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An oriented coloring of planar graphs with girth at least five
Coloring squares of planar graphs with girth six
 SUBMITTED TO EUROPEAN JOURNAL OF COMBINATORICS
"... Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree ∆ ≥ M(g) is (∆+1)colorable. The conjecture is known to be true for g ≥ 7 but false for g ∈ {5, 6}. We show that the conjecture for g = 6 is off ..."
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Cited by 19 (4 self)
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Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree ∆ ≥ M(g) is (∆+1)colorable. The conjecture is known to be true for g ≥ 7 but false for g ∈ {5, 6}. We show that the conjecture for g = 6 is off
Planar graphs of girth at least five . . .
, 2015
"... We prove a conjecture of Dvořák, Král, Nejedlý, and Škrekovski that planar graphs of girth at least five are square ( ∆ + 2)colorable for large enough ∆. In fact, we prove the stronger statement that such graphs are square (∆+2)choosable and even square (∆+2)paintable. ..."
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We prove a conjecture of Dvořák, Král, Nejedlý, and Škrekovski that planar graphs of girth at least five are square ( ∆ + 2)colorable for large enough ∆. In fact, we prove the stronger statement that such graphs are square (∆+2)choosable and even square (∆+2)paintable.
Coloring graphs with fixed genus and girth
 Trans. Am. Math. Soc
, 1997
"... Abstract. It is well known that the maximum chromatic number of a graph on the orientable surface Sg is θ(g1/2). We prove that there are positive constants c1,c2 such that every trianglefree graph on Sg has chromatic number less than c2(g / log(g)) 1/3 and that some trianglefree graph on Sg has ch ..."
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Cited by 25 (1 self)
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Abstract. It is well known that the maximum chromatic number of a graph on the orientable surface Sg is θ(g1/2). We prove that there are positive constants c1,c2 such that every trianglefree graph on Sg has chromatic number less than c2(g / log(g)) 1/3 and that some trianglefree graph on Sg has
On Universal Graphs for Planar Oriented Graphs of a Given Girth
, 1998
"... The oriented chromatic number o(H) of an oriented graph H is dened to be the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . If each graph in a class K has a homomorphism to the same H 0 , then H 0 is Kuniversal. Let P k denote the class of orientations of pla ..."
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Cited by 10 (6 self)
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of planar graphs with girth at least k. Clearly, P 3 P 4 P 5 . . . We discuss the existence of P k universal graphs with special properties. It is known (see [10]) that there exists a P 3 universal graph on 80 vertices. We prove here that (1) there exist no planar P 4 universal graphs; (2
Coloring, sparseness, and girth
, 2015
"... An raugmented tree is a rooted tree plus r edges added from each leaf to ancestors. For d, g, r ∈ N, we construct a bipartite raugmented complete dary tree having girth at least g. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct ..."
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construct sparse nonkchoosable bipartite graphs, showing that maximum average degree at most 2(k − 1) is a sharp sufficient condition for kchoosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of nonkcolorable graphs and hypergraphs with any girth g.
ThreeColoring Klein Bottle Graphs Of Girth Five
 J. COMBIN. THEORY SER. B
, 2004
"... We prove that every graph of girth at least five which admits an embedding in the Klein bottle is 3colorable. This solves a problem raised by Woodburn, and complements a result of Thomassen who proved the same for projective planar and toroidal graphs. ..."
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Cited by 12 (3 self)
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We prove that every graph of girth at least five which admits an embedding in the Klein bottle is 3colorable. This solves a problem raised by Woodburn, and complements a result of Thomassen who proved the same for projective planar and toroidal graphs.
Lucky Choice Number of Planar Graphs with Given Girth
, 2015
"... Suppose the vertices of a graph G are labeled with real numbers. For each vertex v ∈ G, let S(v) denote the sum of the labels of all vertices adjacent to v. A labeling is called lucky if S(u) 6 = S(v) for every pair u and v of adjacent vertices in G. The least integer k for which a graph G has a luc ..."
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strengthening of the results through a list lucky labeling. We apply the discharging method and the Combinatorial Nullstellensatz to show that for a planar graph G of girth at least 26, η(G) ≤ 3. This proves the conjecture for nonbipartite planar graphs of girth at least 26. We also show that for girth
Results 1  10
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1,808