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19
Listcoloring the square of a subcubic graphs
, 2007
"... The square G 2 of a graph G is the graph with the same vertex set as G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that for a planar graph G with maximum degree ∆(G) = 3 we have χ(G 2) ≤ 7. Kostochka and Woodall conjectured that for every graph, the listch ..."
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The square G 2 of a graph G is the graph with the same vertex set as G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that for a planar graph G with maximum degree ∆(G) = 3 we have χ(G 2) ≤ 7. Kostochka and Woodall conjectured that for every graph, the listchromatic number of G 2 equals the chromatic number of G 2, that is χl(G 2) = χ(G 2) for all G. If true, this conjecture (together with Thomassen’s result) implies that every planar graph G with ∆(G) = 3 satisfies χl(G 2) ≤ 7. We prove that every graph (not necessarily planar) with ∆(G) = 3 other than the Petersen graph satisfies χl(G 2) ≤ 8 (and this is best possible). In addition, we show that if G is a planar graph with ∆(G) = 3 and girth g(G) ≥ 7, then χl(G 2) ≤ 7. Dvo˘rák, ˘ Skrekovski, and Tancer showed that if G is a planar graph with ∆(G) = 3 and girth g(G) ≥ 10, then χl(G 2) ≤ 6. We improve the girth bound to show that if G is a planar graph with ∆(G) = 3 and g(G) ≥ 9, then χl(G 2) ≤ 6. All of our proofs can be easily translated into lineartime coloring algorithms.
Injective colorings of planar graphs with few colors
, 2006
"... An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥19 and maximum degree ..."
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An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥19 and maximum degree ∆ are injectively ∆colorable. We also show that all planar graphs of girth ≥ 10 are injectively ( ∆ + 1)colorable, ∆ + 4 colors are sufficient for planar graphs of girth ≥ 5 if ∆ is large enough, and that subcubic planar graphs of girth ≥7 are injectively 5colorable.
Listcolouring squares of sparse subcubic graphs
, 2005
"... The problem of colouring the square of a graph naturally arises in connection with the distance labelings, which have been studied intensively. We consider this problem for sparse subcubic graphs. We show that the choosability χℓ(G2) of the square of a subcubic graph G of maximum average degree d is ..."
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The problem of colouring the square of a graph naturally arises in connection with the distance labelings, which have been studied intensively. We consider this problem for sparse subcubic graphs. We show that the choosability χℓ(G2) of the square of a subcubic graph G of maximum average degree d is at most four if d<24/11 and G does not contain a 5cycle, χℓ(G2)isatmostfiveifd<7/3 and it is at most six if d<5/2. Wegner’s conjecture claims that the chromatic number of the square of a subcubic planar graph is at most seven. Let G be a planar subcubic graph of girth g. Our result implies that χℓ(G2)isatmostfourifg≥24, it is at most 5ifg ≥ 14, and it is at most 6 if g ≥ 10. For lower bounds, we find a planar subcubic graph G1 of girth 9 such that χ(G2 1)=5andaplanar subcubic graph G2 of girth five such that χ(G2 2) = 6. As a consequence, we show that the problem of 4colouring of the square of a subcubic planar graph of girth g = 9 is NPcomplete. We conclude the paper by posing few conjectures.
Frugal Colouring of Graphs
, 2007
"... A kfrugal colouring of a graph G is a proper colouring of the vertices of G such that no colour appears more than k times in the neighbourhood of a vertex. This type of colouring was introduced by Hind, Molloy and Reed in 1997. In this paper, we study the frugal chromatic number of planar graphs, p ..."
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A kfrugal colouring of a graph G is a proper colouring of the vertices of G such that no colour appears more than k times in the neighbourhood of a vertex. This type of colouring was introduced by Hind, Molloy and Reed in 1997. In this paper, we study the frugal chromatic number of planar graphs, planar graphs with large girth, and outerplanar graphs, and relate this parameter with several wellstudied colourings, such as colouring of the square, cyclic colouring, and L(p, q)labelling. We also study frugal edgecolourings of multigraphs.
Distance Constrained Labelings of K4minor Free Graphs
, 2006
"... Motivated by previous results on distance constrained labelings and coloring of squares of K4minor free graphs, we show that for every p> = q> = 1, there exists \Delta 0 such that every K4minor free graph G with maximum degree \Delta> = \Delta 0 has an L(p, q)labeling of span at most qb3 ..."
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Motivated by previous results on distance constrained labelings and coloring of squares of K4minor free graphs, we show that for every p> = q> = 1, there exists \Delta 0 such that every K4minor free graph G with maximum degree \Delta> = \Delta 0 has an L(p, q)labeling of span at most qb3\Delta (G)/2c. The obtained bound is the best possible.
Choosability of the square of a planar graph with maximum degree four, preprint, available at: http://arxiv.org/abs/1303.5156v2
"... Abstract We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree Δ ≤ 4. In particular G 2 is 5, 6, 7, 8choosable if the girth of G is at least 16, 11, 9, 7 respectively. In fact we ..."
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Abstract We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree Δ ≤ 4. In particular G 2 is 5, 6, 7, 8choosable if the girth of G is at least 16, 11, 9, 7 respectively. In fact we prove more general results, in terms of maximum average degree, that imply the results above.
Planar graphs of girth at least five are square (∆ + 2)choosable
, 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.