3,932 research outputs found
First-Fit coloring of Cartesian product graphs and its defining sets
Let the vertices of a Cartesian product graph be ordered by an
ordering . By the First-Fit coloring of we mean the
vertex coloring procedure which scans the vertices according to the ordering
and for each vertex assigns the smallest available color. Let
be the number of colors used in this coloring. By
introducing the concept of descent we obtain a sufficient condition to
determine whether , where and
are arbitrary orders. We study and obtain some bounds for , where is any quasi-lexicographic ordering. The First-Fit
coloring of does not always yield an optimum coloring. A
greedy defining set of is a subset of vertices in the
graph together with a suitable pre-coloring of such that by fixing the
colors of the First-Fit coloring of yields an optimum
coloring. We show that the First-Fit coloring and greedy defining sets of
with respect to any quasi-lexicographic ordering (including the known
lexicographic order) are all the same. We obtain upper and lower bounds for the
smallest cardinality of a greedy defining set in , including some
extremal results for Latin squares.Comment: Accepted for publication in Contributions to Discrete Mathematic
First-Fit coloring of Cartesian product graphs and its defining sets
Let the vertices of a Cartesian product graph be ordered by an ordering . By the First-Fit coloring of we mean the vertex coloring procedure which scans the vertices according to the ordering and for each vertex assigns the smallest available color. Let be the number of colors used in this coloring. By introducing the concept of descent we obtain a sufficient condition to determine whether , where and are arbitrary orders. We study and obtain some bounds for , where is any quasi-lexicographic ordering. The First-Fit coloring of does not always yield an optimum coloring. A greedy defining set of is a subset of vertices in the graph together with a suitable pre-coloring of such that by fixing the colors of the First-Fit coloring of yields an optimum coloring. We show that the First-Fit coloring and greedy defining sets of with respect to any quasi-lexicographic ordering (including the known lexicographic order) are all the same. We obtain upper and lower bounds for the smallest cardinality of a greedy defining set in , including some extremal results for Latin squares
New 2--critical sets in the abelian 2--group
In this paper we determine a class of critical sets in the abelian {2--group}
that may be obtained from a greedy algorithm. These new critical sets are all
2--critical (each entry intersects an intercalate, a trade of size 4) and
completes in a top down manner.Comment: 25 page
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