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The 4choosability of planar graphs without 6cycles
"... Let G be a planar graph without 6cycles. We prove that G is 4choosable. 1 ..."
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Let G be a planar graph without 6cycles. We prove that G is 4choosable. 1
(4, 2)choosability of planar graphs with forbidden structures
, 2015
"... All planar graphs are 4colorable and 5choosable, while some planar graphs are not 4choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any ` ∈ {3 ..."
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` ∈ {3, 4, 5, 6, 7}, a planar graph is 4choosable if it is `cyclefree. In terms of constraining the list assignment, one refinement of kchoosability is choosability with separation. A graph is (k, s)choosable if the graph is colorable from lists of size k where adjacent vertices have at most
Every planar graph without cycles of length 4 to 12 is acyclically 3choosable. LaBRI Research Report RR146209
, 2009
"... An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v ∈ V (G) a list L(v) of available colors. Let G be a graph a ..."
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. In this paper, we prove that every planar graph without cycles of lengths 4 to 12 is acyclically 3choosable. 1
A note on the acyclic 3choosability of some planar graphs
, 2009
"... An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v ∈ V (G) a list L(v) of available colors. Let G be a graph a ..."
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. In this paper, we prove that every planar graph with neither cycles of lengths 4 to 7 (resp. to 8, to 9, to 10) nor triangles at distance less 7 (resp. 5, 3, 2) is acyclically 3choosable.
Acyclic Edge Coloring of Planar Graphs
, 2009
"... An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a ′ (G). It was conjectured by Alon, Sudakov and Zaks ( ..."
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An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a ′ (G). It was conjectured by Alon, Sudakov and Zaks
Acyclic Subgraphs of Planar Digraphs
, 2014
"... An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on n vertices without directed 2cycles possesses an acyclic set of size at least 3n/5. We prove this conjecture for digraphs where every directed cycle has l ..."
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An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on n vertices without directed 2cycles possesses an acyclic set of size at least 3n/5. We prove this conjecture for digraphs where every directed cycle has
Acyclic Edge Colorings of Planar Graphs Without Short Cycles
, 2008
"... A proper edge coloring of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G is the least number of colors in an acyclic edge coloring of G. In this paper, it is proved that the acyclic edge chromatic number of a planar graph G is at most ∆(G) + ..."
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A proper edge coloring of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G is the least number of colors in an acyclic edge coloring of G. In this paper, it is proved that the acyclic edge chromatic number of a planar graph G is at most ∆(G
(1, λ)embedded graphs and the acyclic edge choosability∗
"... A (1, λ)embedded graph is a graph that can be embedded on a surface with Euler characteristic λ so that each edge is crossed by at most one other edge. A graph G is called αlinear if there exists an integral constant β such that e(G′) ≤ αv(G′) + β for each G ′ ⊆ G. In this paper, it is shown tha ..."
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that every (1, λ)embedded graph G is 4linear for all possible λ, and is acyclicly edge(3∆(G) + 70)choosable for λ = 1, 2.
ACYCLIC EDGE COLORING OF PLANAR GRAPHS WITH ∆ COLORS
, 2010
"... Acyclic edge coloring of planar graphs with ∆ colors ..."
Acyclic edge coloring of graphs
, 2013
"... An acyclic edge coloring of a graph G is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The acyclic chromatic index χ′a(G) of a graph G is the least number of colors needed in any acyclic edge colo ..."
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adjacent to cycles of length at most four, with an additional condition that every 5cycle has at most three edges contained in triangles (Theorem 4.4), from which we can conclude some known results as corollaries. We thirdly prove that every planar graph G without intersecting triangles satisfies χ
Results 1  10
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32,211