61 research outputs found
On the multipacking number of grid graphs
In 2001, Erwin introduced broadcast domination in graphs. It is a variant of
classical domination where selected vertices may have different domination
powers. The minimum cost of a dominating broadcast in a graph is denoted
. The dual of this problem is called multipacking: a multipacking
is a set of vertices such that for any vertex and any positive integer
, the ball of radius around contains at most vertices of .
The maximum size of a multipacking in a graph is denoted mp(G). Naturally
mp(G) . Earlier results by Farber and by Lubiw show that
broadcast and multipacking numbers are equal for strongly chordal graphs. In
this paper, we show that all large grids (height at least 4 and width at least
7), which are far from being chordal, have their broadcast and multipacking
numbers equal
Homomorphisms of binary Cayley graphs
A binary Cayley graph is a Cayley graph based on a binary group. In 1982,
Payan proved that any non-bipartite binary Cayley graph must contain a
generalized Mycielski graph of an odd-cycle, implying that such a graph cannot
have chromatic number 3. We strengthen this result first by proving that any
non-bipartite binary Cayley graph must contain a projective cube as a subgraph.
We further conjecture that any homo- morphism of a non-bipartite binary Cayley
graph to a projective cube must be surjective and we prove some special case of
this conjecture
On a class of intersection graphs
Given a directed graph D = (V,A) we define its intersection graph I(D) =
(A,E) to be the graph having A as a node-set and two nodes of I(D) are adjacent
if their corresponding arcs share a common node that is the tail of at least
one of these arcs. We call these graphs facility location graphs since they
arise from the classical uncapacitated facility location problem. In this paper
we show that facility location graphs are hard to recognize and they are easy
to recognize when the graph is triangle-free. We also determine the complexity
of the vertex coloring, the stable set and the facility location problems on
that class
Drawing disconnected graphs on the Klein bottle
We prove that two disjoint graphs must always be drawn separately on the
Klein bottle, in order to minimize the crossing number of the whole drawing.Comment: 13 pages, second version, major changes in the proo
Covering codes in Sierpinski graphs
Graphs and AlgorithmsInternational audienceFor a graph G and integers a and b, an (a, b)-code of G is a set C of vertices such that any vertex from C has exactly a neighbors in C and any vertex not in C has exactly b neighbors in C. In this paper we classify integers a and b for which there exist (a, b)-codes in Sierpinski graphs
Density of -critical signed graphs
We say that a signed graph is -critical if it is not -colorable but
every one of its proper subgraphs is -colorable. Using the definition of
colorability due to Naserasr, Wang, and Zhu that extends the notion of circular
colorability, we prove that every -critical signed graph on vertices has
at least edges, and that this bound is asymptotically tight.
It follows that every signed planar or projective-planar graph of girth at
least is (circular) -colorable, and for the projective-planar case, this
girth condition is best possible. To prove our main result, we reformulate it
in terms of the existence of a homomorphism to the signed graph ,
which is the positive triangle augmented with a negative loop on each vertex.Comment: 27 pages, 12 figure
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