56,248 research outputs found
The chromatic distinguishing index of certain graphs
The distinguishing index of a graph , denoted by , is the least
number of labels in an edge labeling of not preserved by any non-trivial
automorphism. The distinguishing chromatic index of a graph
is the least number such that has a proper edge labeling with
labels that is preserved only by the identity automorphism of . In this
paper we compute the distinguishing chromatic index for some specific graphs.
Also we study the distinguishing chromatic index of corona product and join of
two graphs.Comment: 12 pages, 2 figure
The distinguishing number and the distinguishing index of co-normal product of two graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. The co-normal product
of two graphs and is the graph with vertex set and edge set . In this paper we study the distinguishing number and
the distinguishing index of the co-normal product of two graphs. We prove that
for every , the -th co-normal power of a connected graph with
no false twin vertex and no dominating vertex, has the distinguishing number
and the distinguishing index equal two.Comment: 8 pages. arXiv admin note: text overlap with arXiv:1703.0187
Distinguishing number and distinguishing index of some operations on graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. We examine the effects
on and when is modified by operations on vertex and edge of
. Let be a connected graph of order . We show that , where denotes the graph obtained from by
removal of a vertex and all edges incident to and these inequalities
are true for the distinguishing index. Also we prove that
and , where denotes the graph obtained from
by simply removing the edge . Finally we consider the vertex contraction
and the edge contraction of and prove that the edge contraction decrease
the distinguishing number (index) of by at most one and increase by at most
().Comment: 11 pages, 4 figure
Distinguishing numbers and distinguishing indices of oriented graphs
A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected
graph G is a mapping from the set of vertices (resp. the set of
edges) of G to the set of labels {1,. .. , r} such that no non-trivial
automorphism of G preserves all the vertex (resp. edge) labels. The
distinguishing number D(G) and the distinguishing index D (G) of G are then the
smallest r for which G admits a distinguishing r-vertex-labelling or
r-edge-labelling, respectively. The distinguishing chromatic number D
(G) and the distinguishing chromatic index D (G) are defined similarly,
with the additional requirement that the corresponding labelling must be a
proper colouring. These notions readily extend to oriented graphs, by
considering arcs instead of edges. In this paper, we study the four
corresponding parameters for oriented graphs whose underlying graph is a path,
a cycle, a complete graph or a bipartite complete graph. In each case, we
determine their minimum and maximum value, taken over all possible orientations
of the corresponding underlying graph, except for the minimum values for
unbalanced complete bipartite graphs K m,n with m = 2, 3 or 4 and n > 3, 6 or
13, respectively, or m 5 and n > 2 m -- m 2 , for which we only provide
upper bounds
The distinguishing number and the distinguishing index of line and graphoidal graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. A graphoidal cover of
is a collection of (not necessarily open) paths in such that
every path in has at least two vertices, every vertex of is an
internal vertex of at most one path in and every edge of is in
exactly one path in . Let denote the intersection graph
of . A graph is called a graphoidal graph, if there exists a graph
and a graphoidal cover of such that .
In this paper, we study the distinguishing number and the distinguishing index
of the line graph and the graphoidal graph of a simple connected graph .Comment: 9 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1707.0616
Symmetry breaking in planar and maximal outerplanar graphs
The distinguishing number (index) () of a graph is the
least integer such that has a vertex (edge) labeling with labels
that is preserved only by a trivial automorphism. In this paper we consider the
maximal outerplanar graphs (MOP graphs) and show that MOP graphs, except ,
can be distinguished by at most two vertex (edge) labels. We also compute the
distinguishing number and the distinguishing index of Halin and Mycielskian
graphs.Comment: 10 pages, 6 figure
Distinguishing number and distinguishing index of natural and fractional powers of graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. For any , the -subdivision of is a simple graph
which is constructed by replacing each edge of with a path of length .
The power of , is a graph with same set of vertices of and an
edge between two vertices if and only if there is a path of length at most
between them. The fractional power of , denoted by is
power of the -subdivision of or -subdivision of -th power
of . In this paper we study the distinguishing number and distinguishing
index of natural and fractional powers of . We show that the natural powers
more than two of a graph distinguished by three edge labels. Also we show that
for a connected graph of order with maximum degree , and for , .Comment: 13 page
Distinguishing number and distinguishing index of certain graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. In this paper we
compute these two parameters for some specific graphs. Also we study the
distinguishing number and the distinguishing index of corona product of two
graphs.Comment: 15 pages, 6 figures....To appear in FILOMA
The distinguishing number (index) and the domination number of a graph
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. A set of vertices
in is a dominating set of if every vertex of is
adjacent to some vertex in . The minimum cardinality of a dominating set of
is the domination number of and denoted by . In this paper,
we obtain some upper bounds for the distinguishing number and the
distinguishing index of a graph based on its domination number.Comment: 8 pages, 2 figure
On the neighbour sum distinguishing index of graphs with bounded maximum average degree
A proper edge -colouring of a graph is an assignment of colours to the edges of the graph such that no two
adjacent edges are associated with the same colour. A neighbour sum
distinguishing edge -colouring, or nsd -colouring for short, is a proper
edge -colouring such that for
every edge of . We denote by the neighbour sum
distinguishing index of , which is the least integer such that an nsd
-colouring of exists. By definition at least maximum degree,
colours are needed for this goal. In this paper we prove that for any graph without isolated edges and with , .Comment: 10 page
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