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Circular choosability of graphs
 J. Graph Theory
, 2005
"... This paper discusses the circular version of list coloring of graphs. We give two definitions of the circular list chromatic number (or circular choosability) � �Ð � of a graph � and prove that they are equivalent. Then we prove that for any graph �, � �Ð � � � Ð � . Examples are given to show ..."
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This paper discusses the circular version of list coloring of graphs. We give two definitions of the circular list chromatic number (or circular choosability) � �Ð � of a graph � and prove that they are equivalent. Then we prove that for any graph �, � �Ð � � � Ð � . Examples are given to show that this bound is sharp in the sense that for any ¯ � , there is a graph � with � �Ð � � � Ð � ¯. It is also proved that �degenerate graphs � have � �Ð � � �. This bound is also sharp: for each ¯ � , there is a �degenerate graph � with � �Ð � � � ¯. This shows that � �Ð � could be arbitrarily larger than � Ð �. Finally we prove that if � has maximum degree �, then � �Ð � � �. Key words: circular choosability, circular chromatic number, �degenerate graphs Mathematical Subject Classification: 05C15 1
Planar graphs of oddgirth at least 9 are homomorphic to Petersen graph
"... Let G be a graph and let c: V (G) → � {1,...,5} � 2 be an assignment of 2element subsets of the set {1,...,5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5,2)coloring. A graph is (5,2)colorable if and only if it has a homo ..."
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Cited by 4 (1 self)
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Let G be a graph and let c: V (G) → � {1,...,5} � 2 be an assignment of 2element subsets of the set {1,...,5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5,2)coloring. A graph is (5,2)colorable if and only if it has a homomorphism to the Petersen graph. The oddgirth of a graph G is the length of the shortest odd cycle in G ( ∞ if G is bipartite). We prove that every planar graph of oddgirth at least 9 is (5,2)colorable, and thus it is homomorphic to the Petersen graph. Also, this implies that such graphs have fractional chromatic number at most 5 2. As a special case, this result holds for planar graphs of girth at least 8.
List circular coloring of trees and cycles
 J. Graph Theory
"... DOI 10.1002/jgt.20234 Abstract: SupposeG = (V,E) is a graph and p ≥ 2q are positive integers. A (p,q)coloring of G is a mapping φ: V → {0,1,...,p − 1} such that for any edge xy of G, q ≤ φ(x) − φ(y)  ≤ p − q. A colorlist is a mapping L: V → P({0,1,...,p − 1}) which assigns to each vertex v a s ..."
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DOI 10.1002/jgt.20234 Abstract: SupposeG = (V,E) is a graph and p ≥ 2q are positive integers. A (p,q)coloring of G is a mapping φ: V → {0,1,...,p − 1} such that for any edge xy of G, q ≤ φ(x) − φ(y)  ≤ p − q. A colorlist is a mapping L: V → P({0,1,...,p − 1}) which assigns to each vertex v a set L(v) of permissible colors. An L(p,q)coloring of G is a (p,q)coloring φ of G such that for each vertex v, φ(v) ∈ L(v). We say G is L(p,q)colorable if there exists an L(p,q)coloring ofG. A colorsizelist is a mapping which assigns to each vertex v a nonnegative integer (v). We say G is (p,q)colorable if for every colorlist L with L(v)  = (v), G is L(p,q)colorable. In this article, we consider list circular coloring of trees and cycles. For any tree T and for any p ≥ 2q, we present a necessary and sufficient condition for T to be
(5, 2)Coloring of Sparse Graphs
, 2007
"... We prove that every trianglefree graph whose subgraphs all have average degree less than 12/5 has a (5, 2)coloring. This includes planar and projectiveplanar graphs with girth at least 12. Also, the degree result is sharp; we construct a minimal non(5, 2)colorable trianglefree graph with 10 ve ..."
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We prove that every trianglefree graph whose subgraphs all have average degree less than 12/5 has a (5, 2)coloring. This includes planar and projectiveplanar graphs with girth at least 12. Also, the degree result is sharp; we construct a minimal non(5, 2)colorable trianglefree graph with 10 vertices that has average degree 12/5.
СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ Siberian Electronic Mathematical Reports
"... Abstract. We prove that every trianglefree graph whose subgraphs all have average degree less than 12 5 has a circular (5, 2)coloring. This includes planar and projectiveplanar graphs with girth at least 12. ..."
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Abstract. We prove that every trianglefree graph whose subgraphs all have average degree less than 12 5 has a circular (5, 2)coloring. This includes planar and projectiveplanar graphs with girth at least 12.
Homomorphism Bounded Classes of Graphs
"... Abstract A class C of graphs is said to be Hbounded if each graph in the class C admitsa homomorphism to H. We give a general necessary and sufficient condition forthe existence of bounds with special local properties. This gives a new proof of H"aggkvistHell theorem [5] and implies sever ..."
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Abstract A class C of graphs is said to be Hbounded if each graph in the class C admitsa homomorphism to H. We give a general necessary and sufficient condition forthe existence of bounds with special local properties. This gives a new proof of H&quot;aggkvistHell theorem [5] and implies several cases of the existence of trianglefree bounds for planar graphs.
On Flow and TensionContinuous Maps
, 2002
"... A cycle of a graph G is a set C ⊆ E(G) so that every vertex of the graph (V(G), C) has even degree. If G, H are graphs, we define a map φ: E(G) → E(H) to be cyclecontinuous if the preimage of every cycle of H is a cycle of G. A fascinating conjecture of Jaeger asserts tha ..."
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A cycle of a graph G is a set C &sube; E(G) so that every vertex of the graph (V(G), C) has even degree. If G, H are graphs, we define a map &phi;: E(G) &rarr; E(H) to be cyclecontinuous if the preimage of every cycle of H is a cycle of G. A fascinating conjecture of Jaeger asserts that every bridgeless graph has a cyclecontinuous mapping to the Petersen graph. Jaeger showed that if this conjecture is true, then so is the 5cycledoublecover conjecture and the Fulkerson conjecture. Cycle continuous maps give rise to a natural quasiorder...
Southeast University P.R.China
, 2008
"... This paper considers list circular colouring of graphs in which the colour list assigned to each vertex is an interval of a circle. The circular consecutive choosability chcc(G) of G is defined to be the least t such that for any circle S(r) of length r ≥ χc(G), if each vertex x of G is assigned an ..."
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This paper considers list circular colouring of graphs in which the colour list assigned to each vertex is an interval of a circle. The circular consecutive choosability chcc(G) of G is defined to be the least t such that for any circle S(r) of length r ≥ χc(G), if each vertex x of G is assigned an interval L(x) of S(r) of length t, then there is a circular rcolouring f of G such that f(x) ∈ L(x). We show that for any finite graph G, χ(G) − 1 ≤ chcc(G) < 2χc(G). We determine the value of chcc(G) for complete graphs, trees, even cycles and balanced complete bipartite graphs. Upper and lower bounds for chcc(G) are given for some other classes of graphs. 1
Homomorphism Bounded Classes of Graphs
, 2001
"... A class C of graphs is said to be Hbounded if each graph in the class C admits a homomorphism to H. We give a general necessary and sucient condition for the existence of bounds with special local properties. This gives a new proof of HaggkvistHell theorem [5] and implies several cases of the ..."
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A class C of graphs is said to be Hbounded if each graph in the class C admits a homomorphism to H. We give a general necessary and sucient condition for the existence of bounds with special local properties. This gives a new proof of HaggkvistHell theorem [5] and implies several cases of the existence of triangle free bounds for planar graphs.