14,452 research outputs found
Books versus triangles at the extremal density
A celebrated result of Mantel shows that every graph on vertices with
edges must contain a triangle. A robust version of
this result, due to Rademacher, says that there must in fact be at least
triangles in any such graph. Another strengthening, due
to the combined efforts of many authors starting with Erd\H{o}s, says that any
such graph must have an edge which is contained in at least triangles.
Following Mubayi, we study the interplay between these two results, that is,
between the number of triangles in such graphs and their book number, the
largest number of triangles sharing an edge. Among other results, Mubayi showed
that for any such that any graph
on vertices with at least edges and book number
at most contains at least triangles. He also
asked for a more precise estimate for in terms of . We make a
conjecture about this dependency and prove this conjecture for
and for , thereby answering Mubayi's question in these
ranges.Comment: 15 page
Acyclic edge coloring of graphs
An {\em acyclic edge coloring} of a graph is a proper edge coloring such
that the subgraph induced by any two color classes is a linear forest (an
acyclic graph with maximum degree at most two). The {\em acyclic chromatic
index} \chiup_{a}'(G) of a graph is the least number of colors needed in
an acyclic edge coloring of . Fiam\v{c}\'{i}k (1978) conjectured that
\chiup_{a}'(G) \leq \Delta(G) + 2, where is the maximum degree of
. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC).
A graph with maximum degree at most is {\em
-deletion-minimal} if \chiup_{a}'(G) > \kappa and \chiup_{a}'(H)
\leq \kappa for every proper subgraph of . The purpose of this paper is
to provide many structural lemmas on -deletion-minimal graphs. By using
the structural lemmas, we firstly prove that AECC is true for the graphs with
maximum average degree less than four (\autoref{NMAD4}). We secondly prove that
AECC is true for the planar graphs without triangles adjacent to cycles of
length at most four, with an additional condition that every -cycle has at
most three edges contained in triangles (\autoref{NoAdjacent}), from which we
can conclude some known results as corollaries. We thirdly prove that every
planar graph without intersecting triangles satisfies \chiup_{a}'(G) \leq
\Delta(G) + 3 (\autoref{NoIntersect}). Finally, we consider one extreme case
and prove it: if is a graph with and all the
-vertices are independent, then \chiup_{a}'(G) = \Delta(G). We hope
the structural lemmas will shed some light on the acyclic edge coloring
problems.Comment: 19 page
The bondage number of graphs on topological surfaces and Teschner's conjecture
The bondage number of a graph is the smallest number of its edges whose
removal results in a graph having a larger domination number. We provide
constant upper bounds for the bondage number of graphs on topological surfaces,
improve upper bounds for the bondage number in terms of the maximum vertex
degree and the orientable and non-orientable genera of the graph, and show
tight lower bounds for the number of vertices of graphs 2-cell embeddable on
topological surfaces of a given genus. Also, we provide stronger upper bounds
for graphs with no triangles and graphs with the number of vertices larger than
a certain threshold in terms of the graph genera. This settles Teschner's
Conjecture in positive for almost all graphs.Comment: 21 pages; Original version from January 201
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
How Hard is Counting Triangles in the Streaming Model
The problem of (approximately) counting the number of triangles in a graph is
one of the basic problems in graph theory. In this paper we study the problem
in the streaming model. We study the amount of memory required by a randomized
algorithm to solve this problem. In case the algorithm is allowed one pass over
the stream, we present a best possible lower bound of for graphs
with edges on vertices. If a constant number of passes is allowed,
we show a lower bound of , the number of triangles. We match,
in some sense, this lower bound with a 2-pass -memory algorithm
that solves the problem of distinguishing graphs with no triangles from graphs
with at least triangles. We present a new graph parameter -- the
triangle density, and conjecture that the space complexity of the triangles
problem is . We match this by a second algorithm that solves
the distinguishing problem using -memory
On uniqueness of the q-state Potts model on a self-dual family of graphs
This paper deals with the location of the complex zeros of the Tutte
polynomial for a class of self-dual graphs. For this class of graphs, as the
form of the eigenvalues is known, the regions of the complex plane can be
focused on the sets where there is only one dominant eigenvalue in particular
containing the positive half plane. Thus, in these regions, the analyticity of
the pressure can be derived easily. Next, some examples of graphs with their
Tutte polynomial having a few number of eigenvalues are given. The cases of the
strip of triangles with a double edge, the wheel and the cycle with an edge
having a high order of multiplicity are presented. In particular, for this last
example, we remark that the well known conjecture of Chen et al. is false in
the finite case
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