31 research outputs found
Total coloring of 1-toroidal graphs of maximum degree at least 11 and no adjacent triangles
A {\em total coloring} of a graph is an assignment of colors to the
vertices and the edges of such that every pair of adjacent/incident
elements receive distinct colors. The {\em total chromatic number} of a graph
, denoted by \chiup''(G), is the minimum number of colors in a total
coloring of . The well-known Total Coloring Conjecture (TCC) says that every
graph with maximum degree admits a total coloring with at most colors. A graph is {\em -toroidal} if it can be drawn in torus such
that every edge crosses at most one other edge. In this paper, we investigate
the total coloring of -toroidal graphs, and prove that the TCC holds for the
-toroidal graphs with maximum degree at least~ and some restrictions on
the triangles. Consequently, if is a -toroidal graph with maximum degree
at least~ and without adjacent triangles, then admits a total
coloring with at most colors.Comment: 10 page
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
Recognizing and Drawing IC-planar Graphs
IC-planar graphs are those graphs that admit a drawing where no two crossed
edges share an end-vertex and each edge is crossed at most once. They are a
proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph
with vertices, we present an -time algorithm that computes a
straight-line drawing of in quadratic area, and an -time algorithm
that computes a straight-line drawing of with right-angle crossings in
exponential area. Both these area requirements are worst-case optimal. We also
show that it is NP-complete to test IC-planarity both in the general case and
in the case in which a rotation system is fixed for the input graph.
Furthermore, we describe a polynomial-time algorithm to test whether a set of
matching edges can be added to a triangulated planar graph such that the
resulting graph is IC-planar
On Chromatic Number of Colored Mixed Graphs
An (m, n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f of an (m, n)-colored mixed graph G to an (m, n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f (u)f (v) is an arc (edge) of color c in H. The (m,n)-colored mixed chromatic number x((m,n))(G) of an (m, n)-colored mixed graph G is the order (number of vertices) of a smallest homomorphic image of G. This notion was introduced by Nektfil and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147-155). They showed that x((m,n))(G) <= k(2m n)(k-1) where G is a acyclic k-colorable graph. We prove the tightness of this bound. We also show that the acyclic chromatic number of a graph is bounded by k(2) k(2+) inverted left perpedicular log((2m+n)) log((2m+n)) k inverted right perpendicular if its (m, n)-colored mixed chromatic number is at most k. Furthermore, using probabilistic method, we show that for connected graphs with maximum degree its (m, n)-colored mixed chromatic number is at most 2(Delta-1)(2m+n) (2m + n)(Delta-min(2m+m3)+2) In particular, the last result directly improves the upper bound of 2 Delta(2)2(Delta) oriented chromatic number of graphs with maximum degree Delta, obtained by Kostochka et al. J. Graph Theory 24, 331-340) to 2(Delta-1)(2)2(Delta). We also show that there exists a connected graph with maximum degree Delta and (m, n)-colored mixed chromatic number at least (2m + n)(Delta/2)