30,131 research outputs found

### Normal 6-edge-colorings of some bridgeless cubic graphs

In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of
colors assigned to the edge and the four edges adjacent it, has exactly five or
exactly three distinct colors, respectively. An edge is normal in an
edge-coloring if it is rich or poor in this coloring. A normal
$k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors such
that each edge of the graph is normal. We denote by $\chi'_{N}(G)$ the smallest
$k$, for which $G$ admits a normal $k$-edge-coloring. Normal edge-colorings
were introduced by Jaeger in order to study his well-known Petersen Coloring
Conjecture. It is known that proving $\chi'_{N}(G)\leq 5$ for every bridgeless
cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover,
Jaeger was able to show that it implies classical conjectures like Cycle Double
Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors
were able to show that any simple cubic graph admits a normal
$7$-edge-coloring, and this result is best possible. In the present paper, we
show that any claw-free bridgeless cubic graph, permutation snark, tree-like
snark admits a normal $6$-edge-coloring. Finally, we show that any bridgeless
cubic graph $G$ admits a $6$-edge-coloring such that at least $\frac{7}{9}\cdot
|E|$ edges of $G$ are normal.Comment: 17 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1804.0944

### Impartial coloring games

Coloring games are combinatorial games where the players alternate painting
uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset
specifies that game's coloring constraints. This paper investigates six
impartial rulesets (five new), derived from previously-studied graph coloring
schemes, including proper map coloring, oriented coloring, 2-distance coloring,
weak coloring, and sequential coloring. For each, we study the outcome classes
for special cases and general computational complexity. In some cases we pay
special attention to the Grundy function

### Precoloring co-Meyniel graphs

The pre-coloring extension problem consists, given a graph $G$ and a subset
of nodes to which some colors are already assigned, in finding a coloring of
$G$ with the minimum number of colors which respects the pre-coloring
assignment. This can be reduced to the usual coloring problem on a certain
contracted graph. We prove that pre-coloring extension is polynomial for
complements of Meyniel graphs. We answer a question of Hujter and Tuza by
showing that ``PrExt perfect'' graphs are exactly the co-Meyniel graphs, which
also generalizes results of Hujter and Tuza and of Hertz. Moreover we show
that, given a co-Meyniel graph, the corresponding contracted graph belongs to a
restricted class of perfect graphs (``co-Artemis'' graphs, which are
``co-perfectly contractile'' graphs), whose perfectness is easier to establish
than the strong perfect graph theorem. However, the polynomiality of our
algorithm still depends on the ellipsoid method for coloring perfect graphs

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