894 research outputs found
On incidence coloring conjecture in Cartesian products of graphs
An incidence in a graph is a pair where is a vertex of
and is an edge of incident to . Two incidences and
are adjacent if at least one of the following holds: , , or . An incidence coloring of is a coloring of
its incidences assigning distinct colors to adjacent incidences. It was
conjectured that at most colors are needed for an incidence
coloring of any graph . The conjecture is false in general, but the bound
holds for many classes of graphs. We introduce some sufficient properties of
the two factor graphs of a Cartesian product graph for which admits an
incidence coloring with at most colors.Comment: 11 pages, 5 figure
The Incidence Chromatic Number of Toroidal Grids
An incidence in a graph is a pair with and , such that and are incident. Two incidences and
are adjacent if , or , or the edge equals or . The
incidence chromatic number of is the smallest for which there exists a
mapping from the set of incidences of to a set of colors that assigns
distinct colors to adjacent incidences. In this paper, we prove that the
incidence chromatic number of the toroidal grid equals 5
when and 6 otherwise.Comment: 16 page
On the Structure of the Graph of Unique Symmetric Base Exchanges of Bispanning Graphs
Bispanning graphs are undirected graphs with an edge set that can be
decomposed into two disjoint spanning trees. The operation of symmetrically
swapping two edges between the trees, such that the result is a different pair
of disjoint spanning trees, is called an edge exchange or a symmetric base
exchange. The graph of symmetric base exchanges of a bispanning graph contains
a vertex for every valid pair of disjoint spanning trees, and edges between
them to represent all possible edge exchanges. We are interested in a
restriction of these graphs to only unique symmetric base exchanges, which are
edge exchanges wherein selecting one edge leaves only one choice for selecting
the other. In this thesis, we discuss the structure of the graph of unique
symmetric edge exchanges, and the open question whether these are connected for
all bispanning graphs. Our composition method classifies bispanning graphs by
whether they contain a non-trivial bispanning subgraph, and by vertex- and
edge-connectivity. For bispanning graphs containing a non-trivial bispanning
subgraph, we prove that the unique exchange graph is the Cartesian graph
product of two smaller exchange graphs. For bispanning graphs with
vertex-connectivity two, we show that the bispanning graph is the 2-clique sum
of two smaller bispanning graphs, and that the unique exchange graph can be
built by joining their exchange graphs and forwarding edges at the join seam.
And for all remaining bispanning graphs, we prove a composition method at a
vertex of degree three, wherein the unique exchange graph is constructed from
the exchange graphs of three reduced bispanning graphs.Comment: With minor corrections to v
On the star arboricity of hypercubes
A Hypercube is a graph in which the vertices are all binary vectors of
length n, and two vertices are adjacent if and only if their components differ
in exactly one place. A galaxy or a star forest is a union of vertex disjoint
stars. The star arboricity of a graph , , is the minimum number
of galaxies which partition the edge set of . In this paper among other
results, we determine the exact values of for , . We also improve the last known
upper bound of and show the relation between and
square coloring.Comment: Australas. J. Combin., vol. 59 pt. 2, (2014
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
On the 1-2-3-conjecture
A k-edge-weighting of a graph G is a function w: E(G)->{1,2,...,k}. An
edge-weighting naturally induces a vertex coloring c, where for every vertex v
in V(G), c(v) is sum of weights of the edges that are adjacent to vertex v. If
the induced coloring c is a proper vertex coloring, then w is called a
vertex-coloring k-edge weighting (VCk-EW). Karonski et al. (J. Combin. Theory
Ser. B 91 (2004) 151-157) conjectured that every graph admits a VC3-EW. This
conjecture is known as 1-2-3-conjecture. In this paper, frst, we study the
vertex-coloring edge-weighting of the cartesian product of graphs. Among some
results, we prove that the 1-2-3-conjecture holds for some infinite classes of
graphs. Moreover, we explore some properties of a graph to admit a VC2-EWComment: 13 pages, 3 figure
Strong edge coloring of Cayley graphs and some product graphs
A strong edge coloring of a graph is a proper edge coloring of such
that every color class is an induced matching. The minimum number of colors
required is termed the strong chromatic index. In this paper, we determine the
exact value of the strong chromatic index of all unitary Cayley graphs. Our
investigations reveal an underlying product structure from which the unitary
Cayley graphs emerge. We then go on to give tight bounds for the strong
chromatic index of the Cartesian product of two trees, including an exact
formula for the product in the case of stars. Further, we give bounds for the
strong chromatic index of the product of a tree with a cycle. For any tree,
those bounds may differ from the actual value only by not more than a small
additive constant (at most 2 for even cycles and at most 5 for odd cycles),
moreover they yield the exact value when the length of the cycle is divisible
by
Incidence Choosability of Graphs
An incidence of a graph G is a pair (v, e) where v is a vertex of G and e is
an edge of G incident with v. Two incidences (v, e) and (w, f) of G are
adjacent whenever (i) v = w, or (ii) e = f , or (iii) vw = e or f. An incidence
p-colouring of G is a mapping from the set of incidences of G to the set of
colours {1,. .. , p} such that every two adjacent incidences receive distinct
colours. Incidence colouring has been introduced by Brualdi and Quinn Massey in
1993 and, since then, studied by several authors. In this paper, we introduce
and study the list version of incidence colouring. We determine the exact value
of -- or upper bounds on -- the incidence choice number of several classes of
graphs, namely square grids, Halin graphs, cactuses and Hamiltonian cubic
graphs
On Coupon Colorings of Graphs
Let be a graph with no isolated vertices. A {\em -coupon coloring} of
is an assignment of colors from to the vertices of
such that the neighborhood of every vertex of contains vertices of all
colors from . The maximum for which a -coupon coloring exists is
called the {\em coupon coloring number} of , and is denoted .
In this paper, we prove that every -regular graph has as , and the proportion of
-regular graphs for which tends to
as
Some Cobweb Posets Digraphs' Elementary Properties and Questions
A digraph that represents reasonably a scheduling problem should be a
directed acyclic graph. Here down we shall deal with special kind of graded
named . For their definition and first primary properties see , where natural join of directed biparted graphs and their corresponding
adjacency matrices is defined and then applied to investigate cobweb posets and
their digraphs called . In this report we extend the notion of
cobweb poset while delivering some elementary consequences of the description
and observations established in .Comment: 6 pages, 2 figures,affiliated to The Internet Gian-Carlo Polish
Seminar: http://ii.uwb.edu.pl/akk/sem/sem_rota.ht
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