pinecone/core-2020-05-10-deduplication huggingface dataset:
On the editing distance of graphs

An edge-operation on a graph $G$ is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
$\mathcal{G}$, the editing distance from $G$ to $\mathcal{G}$ is the smallest
number of edge-operations needed to modify $G$ into a graph from $\mathcal{G}$.
In this paper, we fix a graph $H$ and consider ${\rm Forb}(n,H)$, the set of
all graphs on $n$ vertices that have no induced copy of $H$. We provide bounds
for the maximum over all $n$-vertex graphs $G$ of the editing distance from $G$
to ${\rm Forb}(n,H)$, using an invariant we call the {\it binary chromatic
number} of the graph $H$. We give asymptotically tight bounds for that distance
when $H$ is self-complementary and exact results for several small graphs $H$
List colorings with measurable sets

International audienceThe measurable list chromatic number of a graph G is the smallest number such that if each vertex v of G is assigned a set L(v) of measure in a fixed atomless measure space, then there exist sets such that each c(v) has measure one and for every pair of adjacent vertices v and v'. We provide a simpler proof of a measurable generalization of Hall's theorem due to Hilton and Johnson [J Graph Theory 54 (2007), 179-193] and show that the measurable list chromatic number of a finite graph G is equal to its fractional chromatic number
A structure theorem for graphs with no cycle with a unique chord and its
  consequences

We give a structural description of the class $\cal C$ of graphs that do not
contain a cycle with a unique chord as an induced subgraph. Our main theorem
states that any connected graph in $\cal C$ is either in some simple basic
class or has a decomposition. Basic classes are chordless cycles, cliques,
bipartite graphs with one side containing only nodes of degree two and induced
subgraphs of the famous Heawood or Petersen graph. Decompositions are node
cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our
decomposition theorem actually gives a complete structure theorem for ${\cal
C}$, i.e. every graph in ${\cal C}$ can be built from basic graphs that can be
explicitly constructed, and gluing them together by prescribed composition
operations; and all graphs built this way are in ${\cal C}$.
  This has several consequences: an ${\cal O}(nm)$-time algorithm to decide
whether a graph is in $\cal C$, an ${\cal O}(n+m)$-time algorithm that finds a
maximum clique of any graph in $\cal C$ and an ${\cal O}(nm)$-time coloring
algorithm for graphs in $\cal C$. We prove that every graph in $\cal C$ is
either 3-colorable or has a coloring with $\omega$ colors where $\omega$ is the
size of a largest clique. The problem of finding a maximum stable set for a
graph in $\cal C$ is known to be NP-hard
The geometry of t-spreads in k-walk-regular graphs

A graph is walk-regular if the number of closed walks of length\ud
  rooted at a given vertex is a constant through all the vertices for all  .\ud
For a walk-regular graph G with d+1 different eigenvalues and spectrally\ud
maximum diameter D=d, we study the geometry of its d-spreads, that is,\ud
the sets of vertices which are mutually at distance d. When these vertices\ud
are projected onto an eigenspace of its adjacency matrix, we show that\ud
they form a simplex (or tetrahedron in a three-dimensional case) and we\ud
compute its parameters. Moreover, the results are generalized to the case\ud
of k-walk-regular graphs, a family which includes both walk-regular and\ud
distance-regular graphs, and their t-spreads or vertices at distance t from\ud
each other.Preprin
Cubicity of Interval Graphs and the Claw Number

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b-dimensional cube is a Cartesian product l(1) x l(2) x ... x l(b), where each l(i) is a closed interval of unit length on the real line. The cub/city of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b-dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line-i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number psi(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least log(2) psi(G)]. In this article, we show that for an interval graph G log(2) psi(G)-]<= cub(G)<=log(2) psi(G)]+2. It is not clear whether the upper bound of log(2) psi(G)]+2 is tight: till now we are unable to find any interval graph with cub(G)> (log(2)psi(G)]. We also show that for an interval graph G, cub(G) <= log(2) alpha], where alpha is the independence number of G. Therefore, in the special case of psi(G)=alpha, cub(G) is exactly log(2) alpha(2)]. The concept of cubicity can be generalized by considering boxes instead of cubes. A b-dimensional box is a Cartesian product l(1) x l(2) x ... x l(b), where each I is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k-dimensional boxes. It is clear that box(G)<= cub(G). From the above result, it follows that for any graph G, cub(G) <= box(G)log(2) alpha]. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323-333, 201
Facial non-repetitive edge-colouring of plane graphs

International audienceA sequence r1, r2, ..., r2n such that ri=rn+ i for all 1≤i≤n is called a repetition. A sequence S is called non-repetitive if no block (i.e. subsequence of consecutive terms of S) is a repetition. Let G be a graph whose edges are colored. A trail is called non-repetitive if the sequence of colors of its edges is non-repetitive. If G is a plane graph, a facial non-repetitive edge-coloring of G is an edge-coloring such that any facial trail (i.e. a trail of consecutive edges on the boundary walk of a face) is non-repetitive. We denote π′f(G) the minimum number of colors of a facial non-repetitive edge-coloring of G. In this article, we show that π′f(G)≤8 for any plane graph G. We also get better upper bounds for π′f(G) in the cases when G is a tree, a plane triangulation, a simple 3-connected plane graph, a hamiltonian plane graph, an outerplanar graph or a Halin graph. The bound 4 for trees is tight
A Planar Linear Arboricity Conjecture

The linear arboricity la(G) of a graph G is the minimum number of linear
forests that partition the edges of G. In 1984, Akiyama et al. stated the
Linear Arboricity Conjecture (LAC), that the linear arboricity of any simple
graph of maximum degree $\Delta$ is either $\lceil \tfrac{\Delta}{2} \rceil$ or
$\lceil \tfrac{\Delta+1}{2} \rceil$. In [J. L. Wu. On the linear arboricity of
planar graphs. J. Graph Theory, 31:129-134, 1999] and [J. L. Wu and Y. W. Wu.
The linear arboricity of planar graphs of maximum degree seven is four. J.
Graph Theory, 58(3):210-220, 2008.] it was proven that LAC holds for all planar
graphs.
  LAC implies that for $\Delta$ odd, ${\rm la}(G)=\big \lceil \tfrac{\Delta}{2}
\big \rceil$. We conjecture that for planar graphs this equality is true also
for any even $\Delta \ge 6$. In this paper we show that it is true for any even
$\Delta \ge 10$, leaving open only the cases $\Delta=6, 8$.
  We present also an O(n log n)-time algorithm for partitioning a planar graph
into max{la(G),5} linear forests, which is optimal when $\Delta \ge 9$
Multiply Balanced Edge Colorings of Multigraphs

In this paper, a theorem is proved that generalizes several existing
amalgamation results in various ways. The main aim is to disentangle a given
edge-colored amalgamated graph so that the result is a graph in which the edges
are shared out among the vertices in ways that are fair with respect to several
notions of balance (such as between pairs of vertices, degrees of vertices in
the both graph and in each color class, etc). The connectivity of color classes
is also addressed. Most results in the literature on amalgamations focus on the
disentangling of amalgamated complete graphs and complete multipartite graphs.
Many such results follow as immediate corollaries to the main result in this
paper, which addresses amalgamations of graphs in general, allowing for example
the final graph to have multiple edges. A new corollary of the main theorem is
the settling of the existence of Hamilton decompositions of the family of
graphs $K(a_1,\dots, a_p;\lambda_1, \lambda_2)$; such graphs arose naturally in
statistical settings.Comment: 21 pages, 2 figure
Rank-width of Random Graphs

Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs
introduced by Oum and Seymour (2006).
  We investigate the asymptotic behavior of rank-width of a random graph
G(n,p). We show that, asymptotically almost surely, (i) if 0<p<1 is a constant,
then rw(G(n,p)) = \lceil n/3 \rceil-O(1), (ii) if 1/n<< p <1/2, then
rw(G(n,p))= \lceil n/3\rceil-o(n), (iii) if p = c/n and c > 1, then rw(G(n,p))
> r n for some r = r(c), and (iv) if p <= c/n and c<1, then rw(G(n,p)) <=2.
  As a corollary, we deduce that G(n,p) has linear tree-width whenever p=c/n
for each c>1, answering a question of Gao (2006).Comment: 10 page
Excluding induced subdivisions of the bull and related graphs

For any graph $H$, let ${\rm Forb}^*(H)$ be the class of graphs with no
induced subdivision of $H$. It was conjectured in [A.D. Scott, Induced trees in
graphs of large chromatic number, {\em Journal of Graph Theory}, 24:297--311,
1997] that, for every graph $H$, there is a function $f_H:\mathbb{N}
\rightarrow \mathbb{R}$ such that for every graph $G \in {\rm Forb}^*(H)$,
$\chi(G) \leq f_H(\omega(G))$. We prove this conjecture for several graphs $H$,
namely the paw (a triangle with a pendant edge), the bull (a triangle with two
vertex-disjoint pendant edges), and what we call a "necklace," that is, a graph
obtained from a path by choosing a matching such that no edge of the matching
is incident with an endpoint of the path, and for each edge of the matching,
adding a vertex adjacent to the ends of this edge
Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number rc(G) of a connected graph G is the minimum number
of colours needed to colour the edges of G, so that every pair of vertices is
connected by at least one path in which no two edges are coloured the same. In
this paper we show that for every connected graph G, with minimum degree at
least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2,
where {\gamma}_c(G) is the connected domination number of G. Bounds of the form
diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special
graph classes follow as easy corollaries from this result. This includes
interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and
chain graphs all with minimum degree at least 2 and connected. We also show
that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of
these cases, we also demonstrate the tightness of the bounds. An extension of
this idea to two-step dominating sets is used to show that for every connected
graph on n vertices with minimum degree {\delta}, the rainbow connection number
is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of
Schiermeyer (2009), improving the previously best known bound of 20n/{\delta}
by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up
to additive factors by a construction of Caro et al. (2008).Comment: 14 page
Embedding factorizations for 3-uniform hypergraphs

In this paper, two results are obtained on a hypergraph embedding problem.
The proof technique is itself of interest, being the first time amalgamations
have been used to address the embedding of hypergraphs.
  The first result finds necessary and sufficient conditions for the embedding
a hyperedge-colored copy of the complete 3-uniform hypergraph of order $m$,
$K_m^3$, into an $r$-factorization of $K_n^3$, providing that $n>
2m+(-1+\sqrt{8m^2-16m-7})/2$.
  The second result finds necessary and sufficient conditions for an embedding
when not only are the colors of the hyperedges of $K_m^3$ given, but also the
colors of all the "pieces" of hyperedges on these $m$ vertices are prescribed
(the "pieces" of hyperedges are eventually extended to hyperedges of size 3 in
$K_n^3$ by adding new vertices to the hyperedges of size 1 and 2 during the
embedding process).
  Both these results make progress towards settling an old question of Cameron
on completing partial 1-factorizations of hypergraphs.Comment: 8 pages, 1 figur
Vertex coloring of plane graphs with nonrepetitive boundary paths

A sequence $s_1,s_2,...,s_k,s_1,s_2,...,s_k$ is a repetition. A sequence $S$
is nonrepetitive, if no subsequence of consecutive terms of $S$ form a
repetition. Let $G$ be a vertex colored graph. A path of $G$ is nonrepetitive,
if the sequence of colors on its vertices is nonrepetitive. If $G$ is a plane
graph, then a facial nonrepetitive vertex coloring of $G$ is a vertex coloring
such that any facial path is nonrepetitive. Let $\pi_f(G)$ denote the minimum
number of colors of a facial nonrepetitive vertex coloring of $G$. Jendro\vl
and Harant posed a conjecture that $\pi_f(G)$ can be bounded from above by a
constant. We prove that $\pi_f(G)\le 24$ for any plane graph $G$
Linear balanceable and subcubic balanceable graphs

In [{Structural properties and decomposition of linear balanced matrices},
{\it Mathematical Programming}, 55:129--168, 1992], Conforti and Rao
conjectured that every balanced bipartite graph contains an edge that is not
the unique chord of a cycle. We prove this conjecture for balanced bipartite
graphs that do not contain a cycle of length 4 (also known as linear balanced
bipartite graphs), and for balanced bipartite graphs whose maximum degree is at
most 3. We in fact obtain results for more general classes, namely linear
balanceable and subcubic balanceable graphs. Additionally, we prove that cubic
balanced graphs contain a pair of twins, a result that was conjectured by
Morris, Spiga and Webb in [Balanced Cayley graphs and balanced planar graphs,
{\it Discrete Mathematics}, 310:3228--3235, 2010]
On cubic bridgeless graphs whose edge-set cannot be covered by four
  perfect matchings

The problem of establishing the number of perfect matchings necessary to
cover the edge-set of a cubic bridgeless graph is strictly related to a famous
conjecture of Berge and Fulkerson. In this paper we prove that deciding whether
this number is at most 4 for a given cubic bridgeless graph is NP-complete. We
also construct an infinite family $\cal F$ of snarks (cyclically
4-edge-connected cubic graphs of girth at least five and chromatic index four)
whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs
were known. It turns out that the family $\cal F$ also has interesting
properties with respect to the shortest cycle cover problem. The shortest cycle
cover of any cubic bridgeless graph with $m$ edges has length at least
$\tfrac43m$, and we show that this inequality is strict for graphs of $\cal F$.
We also construct the first known snark with no cycle cover of length less than
$\tfrac43m+2$.Comment: 17 pages, 8 figure
On the facial Thue choice index via entropy compression

A sequence is nonrepetitive if it contains no identical consecutive
subsequences. An edge colouring of a path is nonrepetitive if the sequence of
colours of its consecutive edges is nonrepetitive. By the celebrated
construction of Thue, it is possible to generate nonrepetitive edge colourings
for arbitrarily long paths using only three colours. A recent generalization of
this concept implies that we may obtain such colourings even if we are forced
to choose edge colours from any sequence of lists of size 4 (while sufficiency
of lists of size 3 remains an open problem). As an extension of these basic
ideas, Havet, Jendrol', Sot\'ak and \v{S}krabul'\'akov\'a proved that for each
plane graph, 8 colours are sufficient to provide an edge colouring so that
every facial path is nonrepetitively coloured. In this paper we prove that the
same is possible from lists, provided that these have size at least 12. We thus
improve the previous bound of 291 (proved by means of the Lov\'asz Local
Lemma). Our approach is based on the Moser-Tardos entropy-compression method
and its recent extensions by Grytczuk, Kozik and Micek, and by Dujmovi\'c,
Joret, Kozik and Wood
1-factor and cycle covers of cubic graphs

Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$.
Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$
1-factors. Let $\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$
1-factors of $G$.
  Any list of three 1-factors induces a core of a cubic graph. We use results
on the structure of cores to prove sufficient conditions for Berge-covers and
for the existence of three 1-factors with empty intersection. Furthermore, if
$\mu_3(G) \not = 0$, then $2 \mu_3(G)$ is an upper bound for the girth of $G$.
We also prove some new upper bounds for the length of shortest cycle covers of
bridgeless cubic graphs.
  Cubic graphs with $\mu_4(G) = 0$ have a 4-cycle cover of length $\frac{4}{3}
|E(G)|$ and a 5-cycle double cover. These graphs also satisfy two conjectures
of Zhang. We also give a negative answer to a problem of Zhang.Comment: final versio
Decomposing graphs of high minimum degree into 4-cycles

If a graph G decomposes into edge-disjoint 4-cycles, then each vertex of G has even degree and 4 divides the number of edges in G. It is shown that these obvious necessary conditions are also sufficient when G is any simple graph having minimum degree at least (3132+on(1))n, where n is the number of vertices in G. This improves the bound given by Gustavsson (PhD Thesis, University of Stockholm, 1991), who showed (as part of a more general result) sufficiency for simple graphs with minimum degree at least (1-10-94+on(1))n. On the other hand, it is known that for arbitrarily large values of n there exist simple graphs satisfying the obvious necessary conditions, having n vertices and minimum degree 35n-1, but having no decomposition into edge-disjoint 4-cycles. We also show that if G is a bipartite simple graph with n vertices in each part, then the obvious necessary conditions for G to decompose into 4-cycles are sufficient when G has minimum degree at least (3132+on(1))n
Circumference and Pathwidth of Highly Connected Graphs

Birmele [J. Graph Theory, 2003] proved that every graph with circumference t
has treewidth at most t-1. Under the additional assumption of 2-connectivity,
such graphs have bounded pathwidth, which is a qualitatively stronger result.
Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007]
who showed that every graph without k disjoint cycles of length at least t has
bounded treewidth (as a function of k and t). Our main result states that,
under the additional assumption of (k + 1)- connectivity, such graphs have
bounded pathwidth. In fact, they have pathwidth O(t^3 + tk^2). Moreover,
examples show that (k + 1)-connectivity is required for bounded pathwidth to
hold. These results suggest the following general question: for which values of
k and graphs H does every k-connected H-minor-free graph have bounded
pathwidth? We discuss this question and provide a few observations.Comment: 11 pages, 4 figure
On a Conjecture of Erd\H{o}s, Gallai, and Tuza

Erd\H{o}s, Gallai, and Tuza posed the following problem: given an $n$-vertex
graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose
deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size
of a set of edges containing at most one edge from each triangle of $G$. Is it
always the case that $\alpha_1(G) + \tau_1(G) \leq n^2/4$? We have two main
results. We first obtain the upper bound $\alpha_1(G) + \tau_1(G) \leq
5n^2/16$, as a partial result towards the Erd\H{o}s--Gallai--Tuza conjecture.
We also show that always $\alpha_1(G) \leq n^2/2 - m$, where $m$ is the number
of edges in $G$; this bound is sharp in several notable cases.Comment: 5 pages, minor revisions: added new details, new conjecture, and
  cleaned up notation slightl
Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.

We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time
Relations between the local chromatic number and its directed version

The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here, we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph. We show that for fractional versions the analogous problem has a different answer: there always exists an orientation for which the directed and undirected values coincide. We also determine the supremum of the possible ratios of these fractional parameters, which turns out to be e, the basis of the natural logarithm
On the structure of graphs with given odd girth and large minimum degree

We study minimum degree conditions for which a graph with given odd girth has
a simple structure. For example, the classical work of Andr\'asfai, Erd\H os,
and S\'os implies that every $n$-vertex graph with odd girth $2k+1$ and minimum
degree bigger than $\frac{2}{2k+1}n$ must be bipartite. We consider graphs with
a weaker condition on the minimum degree. Generalizing results of H\"aggkvist
and of H\"aggkvist and Jin for the cases $k=2$ and $3$, we show that every
$n$-vertex graph with odd girth $2k+1$ and minimum degree bigger than
$\frac{3}{4k}n$ is homomorphic to the cycle of length $2k+1$. This is best
possible in the sense that there are graphs with minimum degree $\frac{3}{4k}n$
and odd girth $2k+1$ which are not homomorphic to the cycle of length $2k+1$.
Similar results were obtained by Brandt and Ribe-Baumann