11,693 research outputs found
Edge-coloring of split graphs
Orientador: Célia Picinin de MelloTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Por apresentar basicamente fórmulas, o resumo, na íntegra, poderá ser visualizado no texto completo da tese digitalAbstract: Not informedDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã
Split-critical and uniquely split-colorable graphs
The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some properties of split-critical and uniquely split-colorable graphs. We describe constructions of such graphs with some additional properties. We also study the effect of the addition and the removal of some edge sets on the value of the split-chromatic number. All these results are compared with their cochromatic counterparts. We conclude with several research directions on the topic
New Results on Edge-coloring and Total-coloring of Split Graphs
A split graph is a graph whose vertex set can be partitioned into a clique
and an independent set. A connected graph is said to be -admissible if
admits a special spanning tree in which the distance between any two adjacent
vertices is at most . Given a graph , determining the smallest for
which is -admissible, i.e. the stretch index of denoted by
, is the goal of the -admissibility problem. Split graphs are
-admissible and can be partitioned into three subclasses: split graphs with
or . In this work we consider such a partition while dealing
with the problem of coloring a split graph. Vizing proved that any graph can
have its edges colored with or colors, and thus can be
classified as Class 1 or Class 2, respectively. When both, edges and vertices,
are simultaneously colored, i.e., a total coloring of , it is conjectured
that any graph can be total colored with or colors, and
thus can be classified as Type 1 or Type 2. These both variants are still open
for split graphs. In this paper, using the partition of split graphs presented
above, we consider the edge coloring problem and the total coloring problem for
split graphs with . For this class, we characterize Class 2 and Type
2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type
1 graph.Comment: 20 pages, 5 figure
Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs
A path in an edge-colored graph is rainbow if no two edges of it are
colored the same. The graph is rainbow-connected if there is a rainbow path
between every pair of vertices. If there is a rainbow shortest path between
every pair of vertices, the graph is strongly rainbow-connected. The
minimum number of colors needed to make rainbow-connected is known as the
rainbow connection number of , and is denoted by . Similarly,
the minimum number of colors needed to make strongly rainbow-connected is
known as the strong rainbow connection number of , and is denoted by
. We prove that for every , deciding whether
is NP-complete for split graphs, which form a subclass
of chordal graphs. Furthermore, there exists no polynomial-time algorithm for
approximating the strong rainbow connection number of an -vertex split graph
with a factor of for any unless P = NP. We
then turn our attention to block graphs, which also form a subclass of chordal
graphs. We determine the strong rainbow connection number of block graphs, and
show it can be computed in linear time. Finally, we provide a polynomial-time
characterization of bridgeless block graphs with rainbow connection number at
most 4.Comment: 13 pages, 3 figure
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