259 research outputs found
Extremal Directed And Mixed Graphs
We consider three problems in extremal graph theory, namely the degree/diameter problem, the degree/geodecity problem and Tur\'{a}n problems, in the context of directed and partially directed graphs.
A directed graph or mixed graph is -geodetic if there is no pair of vertices of such that there exist distinct non-backtracking walks with length in from to . The order of a -geodetic digraph with minimum out-degree is bounded below by the \emph{directed Moore bound} ; similarly the order of a -geodetic mixed graph with minimum undirected degree and minimum directed out-degree is bounded below by the \emph{mixed Moore bound}. We will be interested in networks with order exceeding the Moore bound by some small \emph{excess} .
The \emph{degree/geodecity problem} asks for the smallest possible order of a -geodetic digraph or mixed graph with given degree parameters. We prove the existence of extremal graphs, which we call \emph{geodetic cages}, and provide some bounds on their order and information on their structure.
We discuss the structure of digraphs with excess one and rule out the existence of certain digraphs with excess one. We then classify all digraphs with out-degree two and excess two, as well as all diregular digraphs with out-degree two and excess three. We also present the first known non-trivial examples of geodetic cages.
We then generalise this work to the setting of mixed graphs. First we address the question of the total regularity of mixed graphs with order close to the Moore bound and prove bounds on the order of mixed graphs that are not totally regular. In particular using spectral methods we prove a conjecture of L\'{o}pez and Miret that mixed graphs with diameter two and order one less than the Moore bound are totally regular.
Using counting arguments we then provide strong bounds on the order of totally regular -geodetic mixed graphs and use these results to derive new extremal mixed graphs.
Finally we change our focus and study the Tur\'{a}n problem of the largest possible size of a -geodetic digraph with given order. We solve this problem and also prove an exact expression for the restricted problem of the largest possible size of strongly connected -geodetic digraphs, as well as providing constructions of strongly connected -geodetic digraphs that we conjecture to be extremal for larger . We close with a discussion of some related generalised Tur\'{a}n problems for -geodetic digraphs
Algebraic Connectivity and Degree Sequences of Trees
We investigate the structure of trees that have minimal algebraic
connectivity among all trees with a given degree sequence. We show that such
trees are caterpillars and that the vertex degrees are non-decreasing on every
path on non-pendant vertices starting at the characteristic set of the Fiedler
vector.Comment: 8 page
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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