101 research outputs found
The linear arboricity conjecture for graphs of low degeneracy
A -linear coloring of a graph is an edge coloring of with
colors so that each color class forms a linear forest -- a forest whose each
connected component is a path. The linear arboricity of is the
minimum integer such that there exists a -linear coloring of .
Akiyama, Exoo and Harary conjectured in 1980 that for every graph ,
where
is the maximum degree of . First, we prove the conjecture for
3-degenerate graphs. This establishes the conjecture for graphs of treewidth at
most 3 and provides an alternative proof for the conjecture in some classes of
graphs like cubic graphs and triangle-free planar graphs for which the
conjecture was already known to be true. Next, for every 2-degenerate graph
, we show that if
. We conjecture that this equality holds also when
and show that this is the case for some well-known
subclasses of 2-degenerate graphs. All our proofs can be converted into linear
time algorithms.Comment: 23 pages, 6 figures, preliminary version appeared in the proceedings
of WG 202
Equitable partition of graphs into induced forests
An equitable partition of a graph is a partition of the vertex-set of
such that the sizes of any two parts differ by at most one. We show that every
graph with an acyclic coloring with at most colors can be equitably
partitioned into induced forests. We also prove that for any integers
and , any -degenerate graph can be equitably
partitioned into induced forests.
Each of these results implies the existence of a constant such that for
any , any planar graph has an equitable partition into induced
forests. This was conjectured by Wu, Zhang, and Li in 2013.Comment: 4 pages, final versio
Separation dimension of bounded degree graphs
The 'separation dimension' of a graph is the smallest natural number
for which the vertices of can be embedded in such that any
pair of disjoint edges in can be separated by a hyperplane normal to one of
the axes. Equivalently, it is the smallest possible cardinality of a family
of total orders of the vertices of such that for any two
disjoint edges of , there exists at least one total order in
in which all the vertices in one edge precede those in the other. In general,
the maximum separation dimension of a graph on vertices is . In this article, we focus on bounded degree graphs and show that the
separation dimension of a graph with maximum degree is at most
. We also demonstrate that the above bound is nearly
tight by showing that, for every , almost all -regular graphs have
separation dimension at least .Comment: One result proved in this paper is also present in arXiv:1212.675
Linear arboricity of degenerate graphs
A linear forest is a union of vertex-disjoint paths, and the linear
arboricity of a graph , denoted by , is the minimum
number of linear forests needed to partition the edge set of . Clearly,
for a graph with maximum
degree . On the other hand, the Linear Arboricity Conjecture due to
Akiyama, Exoo, and Harary from 1981 asserts that for every graph . This conjecture has been
verified for planar graphs and graphs whose maximum degree is at most , or
is equal to or .
Given a positive integer , a graph is -degenerate if it can be
reduced to a trivial graph by successive removal of vertices with degree at
most . We prove that for any -degenerate graph , provided .Comment: 15 pages, 1 figur
Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six
We consider the problem of counting all k-vertex subgraphs in an input graph, for any constant k. This problem (denoted SUB-CNT_k) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for SUB-CNT_k. Towards a better understanding of the limits of these techniques, we ask: for what values of k can SUB_CNT_k be solved in linear time?
We discover a chasm at k=6. Specifically, we prove that for k < 6, SUB_CNT_k can be solved in linear time. Assuming a standard conjecture in fine-grained complexity, we prove that for all k ? 6, SUB-CNT_k cannot be solved even in near-linear time
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