pinecone/core-2020-05-10-deduplication huggingface dataset:
Cores of imprimitive symmetric graphs of order a product of two distinct
  primes

A retract of a graph $\Gamma$ is an induced subgraph $\Psi$ of $\Gamma$ such
that there exists a homomorphism from $\Gamma$ to $\Psi$ whose restriction to
$\Psi$ is the identity map. A graph is a core if it has no nontrivial retracts.
In general, the minimal retracts of a graph are cores and are unique up to
isomorphism; they are called the core of the graph. A graph $\Gamma$ is
$G$-symmetric if $G$ is a subgroup of the automorphism group of $\Gamma$ that
is transitive on the vertex set and also transitive on the set of ordered pairs
of adjacent vertices. If in addition the vertex set of $\Gamma$ admits a
nontrivial partition that is preserved by $G$, then $\Gamma$ is an imprimitive
$G$-symmetric graph. In this paper cores of imprimitive symmetric graphs
$\Gamma$ of order a product of two distinct primes are studied. In many cases
the core of $\Gamma$ is determined completely. In other cases it is proved that
either $\Gamma$ is a core or its core is isomorphic to one of two graphs, and
conditions on when each of these possibilities occurs is given
List Colorings with Distinct List Sizes, the Case of Complete Bipartite Graphs

Let f:V→N be a function on the vertex set of the graph G=(V,E). The graph G is f-choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number, χsc(G), is the minimum of ∑f(v), over all functions f such that G is f-choosable. It is known (Alon, Surveys in Combinatorics, 1993 (Keele), London Mathematical Society Lecture Note Series, Vol. 187, Cambridge University Press, Cambridge, 1993, pp. 1-33, Random Struct Algor 16 (2000), 364-368) that if G has average degree d, then the usual choice number χℓ(G) is at least Ω(logd), so they grow simultaneously. In this article, we show that χsc(G)/|V(G)| can be bounded while the minimum degree δmin(G)→∞. Our main tool is to give tight estimates for the sum choice number of the unbalanced complete bipartite graph Ka,q. © 2015 Wiley Periodicals, Inc
Ramsey number of a connected triangle matching

We determine the $2$-color Ramsey number of a {\em connected} triangle
matching $c(nK_3)$ which is any connected graph containing $n$ vertex disjoint
triangles. We obtain that $R(c(nK_3),c(nK_3))=7n-2$, somewhat larger than in
the classical result of Burr, Erd\H os and Spencer for a triangle matching,
$R(nK_3,nK_3)=5n$. The motivation is to determine the Ramsey number
$R(C_n^2,C_n^2)$ of the square of a cycle $C_n^2$. We apply our Ramsey result
for connected triangle matchings to show that the Ramsey number of an "almost"
square of a cycle $C_n^{2,c}$ (a cycle of length $n$ in which all but at most a
constant number $c$ of short diagonals are present) is asymptotic to $7n/3$.Comment: Journal of Graph Theory, 201
Connectedness and isomorphism properties of the zig-zag product of
  graphs

In this paper we investigate the connectedness and the isomorphism problems
for zig-zag products of two graphs. A sufficient condition for the zig-zag
product of two graphs to be connected is provided, reducing to the study of the
connectedness property of a new graph which depends only on the second factor
of the graph product. We show that, when the second factor is a cycle graph,
the study of the isomorphism problem for the zig-zag product is equivalent to
the study of the same problem for the associated pseudo-replacement graph. The
latter is defined in a natural way, by a construction generalizing the
classical replacement product, and its degree is smaller than the degree of the
zig-zag product graph.
  Two particular classes of products are studied in detail: the zig-zag product
of a complete graph with a cycle graph, and the zig-zag product of a
$4$-regular graph with the cycle graph of length $4$. Furthermore, an example
coming from the theory of Schreier graphs associated with the action of
self-similar groups is also considered: the graph products are completely
determined and their spectral analysis is developed.Comment: 33 page
A reconfigurations analogue of Brooks' Theorem and its consequences.

Let G be a simple undirected connected graph on n vertices with maximum degree Δ. Brooks' Theorem states that G has a proper Δ-coloring unless G is a complete graph, or a cycle with an odd number of vertices. To recolor G is to obtain a new proper coloring by changing the color of one vertex. We show an analogue of Brooks' Theorem by proving that from any k-coloring, inline image, a Δ-coloring of G can be obtained by a sequence of inline image recolorings using only the original k colors unless\ud
\ud
–\ud
G is a complete graph or a cycle with an odd number of vertices, or\ud
–\ud
inline image, G is Δ-regular and, for each vertex v in G, no two neighbors of v are colored alike.\ud
We use this result to study the reconfiguration graph inline image of the k-colorings of G. The vertex set of inline image is the set of all possible k-colorings of G and two colorings are adjacent if they differ on exactly one vertex. We prove that for inline image, inline image consists of isolated vertices and at most one further component that has diameter inline image. This result enables us to complete both a structural and an algorithmic characterization for reconfigurations of colorings of graphs of bounded maximum degree
Edmonds' Branching Theorem in Digraphs without Forward-infinite Paths

Let $ D $ be a finite digraph, and let $ V_0,\dots,V_{k-1} $ be nonempty
subsets of $ V(D) $. The (strong form of) Edmonds' branching theorem states
thatthere are pairwise edge-disjoint spanning branchings $ \mathcal{B}_0,\dots,
\mathcal{B}_{k-1} $ in $ D $ such that the root set of $ \mathcal{B}_i $ is $
V_i\ (i=0,\dots,k-1) $ if and only if for all $ \varnothing \neq X\subseteq
V(D) $ the number of ingoing edges of $ X $ is greater than or equal to the
number of sets $ V_i $ disjoint from $ X $. As was shown by R. Aharoni and C.
Thomassen in \cite{aharoni1989infinite}, this theorem does not remain true for
infinite digraphs. Thomassen also proved that for the class of digraphs without
backward-infinite paths, the above theorem of Edmonds remains true. Our main
result is that for digraphs without forward-infinite paths, Edmonds' branching
theorem remains true as well
Some Remarks on Rainbow Connectivity

An edge (vertex) coloured graph is rainbow-connected if there is a rainbow
path between any two vertices, i.e. a path all of whose edges (internal
vertices) carry distinct colours. Rainbow edge (vertex) connectivity of a graph
$G$ is the smallest number of colours needed for a rainbow edge (vertex)
colouring of $G$. In this paper we propose a very simple approach to studying
rainbow connectivity in graphs. Using this idea, we give a unified proof of
several known results, as well as some new ones
List colouring with a bounded palette

Kr\'al' and Sgall (2005) introduced a refinement of list colouring where
every colour list must be subset to one predetermined palette of colours. We
call this $(k,\ell)$-choosability when the palette is of size at most $\ell$
and the lists must be of size at least $k$. They showed that, for any integer
$k\ge 2$, there is an integer $C=C(k,2k-1)$, satisfying $C = O(16^{k}\ln k)$ as
$k\to \infty$, such that, if a graph is $(k,2k-1)$-choosable, then it is
$C$-choosable, and asked if $C$ is required to be exponential in $k$. We
demonstrate it must satisfy $C = \Omega(4^k/\sqrt{k})$.
  For an integer $\ell \ge 2k-1$, if $C(k,\ell)$ is the least integer such that
a graph is $C(k,\ell)$-choosable if it is $(k,\ell)$-choosable, then we more
generally supply a lower bound on $C(k,\ell)$, one that is super-polynomial in
$k$ if $\ell = o(k^2/\ln k)$, by relation to an extremal set theoretic
property. By the use of containers, we also give upper bounds on $C(k,\ell)$
that improve on earlier bounds if $\ell \ge 2.75 k$.Comment: 12 pages, 1 figure, 1 table; to appear in Journal of Graph Theor
Petersen cores and the oddness 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 $G$. If $G$ is not 3-edge-colorable, then $\mu_3(G) \geq 3$. In
[E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78(3)
(2015) 195-206] it is shown that if $\mu_3(G) \not = 0$, then $2 \mu_3(G)$ is
an upper bound for the girth of $G$. We show that $\mu_3(G)$ bounds the oddness
$\omega(G)$ of $G$ as well. We prove that $\omega(G)\leq \frac{2}{3}\mu_3(G)$.
If $\mu_3(G) = \frac{2}{3} \mu_3(G)$, then every $\mu_3(G)$-core has a very
specific structure. We call these cores Petersen cores. We show that for any
given oddness there is a cyclically 4-edge-connected cubic graph $G$ with
$\omega(G) = \frac{2}{3}\mu_3(G)$. On the other hand, the difference between
$\omega(G)$ and $\frac{2}{3}\mu_3(G)$ can be arbitrarily big. This is true even
if we additionally fix the oddness. Furthermore, for every integer $k\geq 3$,
there exists a bridgeless cubic graph $G$ such that $\mu_3(G)=k$.Comment: 13 pages, 9 figure
A survey on the computational complexity of coloring graphs with forbidden subgraphs.

For a positive integer k, a k-coloring of a graph inline image is a mapping inline image such that inline image whenever inline image. The COLORING problem is to decide, for a given G and k, whether a k-coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k-COLORING instead. We survey known results on the computational complexity of COLORING and k-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex
Calculating Ramsey numbers by partitioning colored graphs

In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k at least 1, in every edge colouring of a complete graph with the colours red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete (k+1)-partite graph. When the colouring is connected in red, we prove a stronger result - that it is possible to cover all the vertices with k red paths and a blue balanced complete (k+2)-partite graph. \ud
Using these results we determine the Ramsey number of a path on n vertices, versus a balanced complete k-partite graph, with m vertices in each part, whenever m-1 is divisible by n-1. This generalizes a result of Erdos who proved the m=1 case of this result. We also determine the Ramsey number of a path on n vertices versus the power of a path on n vertices. This solves a conjecture of Allen, Brightwell, and Skokan
On the number of k-dominating independent sets

We study the existence and the number of $k$-dominating independent sets in
certain graph families. While the case $k=1$ namely the case of maximal
independent sets - which is originated from Erd\H{o}s and Moser - is widely
investigated, much less is known in general. In this paper we settle the
question for trees and prove that the maximum number of $k$-dominating
independent sets in $n$-vertex graphs is between $c_k\cdot\sqrt[2k]{2}^n$ and
$c_k'\cdot\sqrt[k+1]{2}^n$ if $k\geq 2$, moreover the maximum number of
$2$-dominating independent sets in $n$-vertex graphs is between $c\cdot 1.22^n$
and $c'\cdot1.246^n$. Graph constructions containing a large number of
$k$-dominating independent sets are coming from product graphs, complete
bipartite graphs and with finite geometries. The product graph construction is
associated with the number of certain MDS codes.Comment: 13 page
Expression for the Number of Spanning Trees of Line Graphs of Arbitrary
  Connected Graphs

For any graph $G$, let $t(G)$ be the number of spanning trees of $G$, $L(G)$
be the line graph of $G$ and for any non-negative integer $r$, $S_r(G)$ be the
graph obtained from $G$ by replacing each edge $e$ by a path of length $r+1$
connecting the two ends of $e$. In this paper we obtain an expression for
$t(L(S_r(G)))$ in terms of spanning trees of $G$ by a combinatorial approach.
This result generalizes some known results on the relation between
$t(L(S_r(G)))$ and $t(G)$ and gives an explicit expression
$t(L(S_r(G)))=k^{m+s-n-1}(rk+2)^{m-n+1}t(G)$ if $G$ is of order $n+s$ and size
$m+s$ in which $s$ vertices are of degree $1$ and the others are of degree $k$.
Thus we prove a conjecture on $t(L(S_1(G)))$ for such a graph $G$.Comment: 22 pages, 7 pages, presented at the National Conference of
  Combinatorics and Graph Theory at Guangzhou, China in Nov. 2014. It was
  submitted to JGT in Feb. 2014 and revised in July 201
Partitioning Perfect Graphs into Stars

The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
A note on the order of iterated line digraphs

Given a digraph G, we propose a new method to find the recurrence equation for the number of vertices n_k of the k-iterated line digraph L_k(G), for k>= 0, where L_0(G) = G. We obtain this result by using the minimal\ud
polynomial of a quotient digraph pi(G) of G. We show some examples of this method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic digraphs. In the first case, our method gives the enumeration of the ternary length-2 squarefree words of any length.Peer ReviewedPostprint (author's final draft
On (4,2)-Choosable Graphs

A graph $G$ is called $(a,b)$-choosable if for any list assignment $L$ which
assigns to each vertex $v$ a set $L(v)$ of $a$ permissible colours, there is a
$b$-tuple $L$-colouring of $G$. An $(a,1)$-choosable graph is also called
$a$-choosable. In the pioneering paper on list colouring of graphs by
Erd\H{o}s, Rubin and Taylor, $2$-choosable graphs are characterized. Confirming
a special case of a conjecture of Erd\H{o}s--Rubin--Taylor, Tuza and Voigt
proved that $2$-choosable graphs are $(2m,m)$-choosable for any positive
integer $m$. On the other hand, Voigt proved that if $m$ is an odd integer,
then these are the only $(2m,m)$-choosable graphs; however, when $m$ is even,
there are $(2m,m)$-choosable graphs that are not $2$-choosable. A graph is
called $3$-choosable-critical if it is not $2$-choosable, but all its proper
subgraphs are $2$-choosable. Voigt conjectured that for every positive integer
$m$, all bipartite $3$-choosable-critical graphs are $(4m,2m)$-choosable. In
this paper, we determine which $3$-choosable-critical graphs are
$(4,2)$-choosable, refuting Voigt's conjecture in the process. Nevertheless, a
weaker version of the conjecture is true: we prove that there is an even
integer $k$ such that for any positive integer $m$, every bipartite
$3$-choosable-critical graph is $(2km,km)$-choosable. Moving beyond
$3$-choosable-critical graphs, we present an infinite family of
non-$3$-choosable-critical graphs which have been shown by computer analysis to
be $(4,2)$-choosable. This shows that the family of all $(4,2)$-choosable
graphs has rich structure.Comment: 18 pages, 8 figures. Now includes source code in ancillary file
On the Rigidity of Sparse Random Graphs

A graph with a trivial automorphism group is said to be rigid. Wright proved
that for $\frac{\log n}{n}+\omega(\frac 1n)\leq p\leq \frac 12$ a random graph
$G\in G(n,p)$ is rigid whp. It is not hard to see that this lower bound is
sharp and for $p<\frac{(1-\epsilon)\log n}{n}$ with positive probability
$\text{aut}(G)$ is nontrivial. We show that in the sparser case $\omega(\frac 1
n)\leq p\leq \frac{\log n}{n}+\omega(\frac 1n)$, it holds whp that $G$'s
$2$-core is rigid. We conclude that for all $p$, a graph in $G(n,p)$ is
reconstrutible whp. In addition this yields for $\omega(\frac 1n)\leq p\leq
\frac 12$ a canonical labeling algorithm that almost surely runs in polynomial
time with $o(1)$ error rate. This extends the range for which such an algorithm
is currently known.Comment: 17 page
A Density Tur\'an Theorem

Let $F$ be a graph which contains an edge whose deletion reduces its
chromatic number. For such a graph $F,$ a classical result of Simonovits from
1966 shows that every graph on $n\ge n_0(F)$ vertices with more than
$\frac{\chi(F)-2}{\chi(F)-1}\cdot \frac{n^2}{2}$ edges contains a copy of $F$.
In this paper we derive a similar theorem for multipartite graphs.
  For a graph $H$ and an integer $\ell \geq v(H)$, let $d_{\ell}(H)$ be the
minimum real number such that every $\ell$-partite graph whose edge density
between any two parts is greater than $d_{\ell}(H)$ contains a copy of $H$. Our
main contribution is to show that $d_{\ell}(H)=\frac{\chi(H)-2}{\chi(H)-1}$ for
$\ell \ge \ell_0(H)$ sufficiently large if and only if $H$ admits a
vertex-colouring with $\chi(H)-1$ colours such that all colour classes but one
are independent sets, and the exceptional class induces just a matching. When
$H$ is a clique, this recovers a result of Pfender [Complete subgraphs in
multipartite graphs, Combinatorica 32 (2012), 483--495]. We also consider
several extensions of Pfender's result.Comment: 28 pages, 2 figure
Existence of μ-representation of graphs

Recently, Jones et al. introduced the study of μ-representable graphs, where μ is a word over { 1,2} containing at least one 1. The notion of a μ-representable graph is a far-reaching generalization of the notion of a word-representable graph studied in the literature in a series of papers. Jones et al. have shown that any graph is 11⋯1-representable assuming that the number of 1s is at least three, while the class of 12-rerpesentable graphs is properly contained in the class of comparability graphs, which, in turn, is properly contained in the class of word-representable graphs corresponding to 11-representable graphs. Further studies in this direction were conducted by Nabawanda, who has shown, in particular, that the class of 112-representable graphs is not included in the class of word-representable graphs. Jones et al. raised a question on classification of μ-representable graphs at least for small values of μ. In this paper we show that if μ is of length at least 3 then any graph is μ-representable. This rather unexpected result shows that from existence of representation point of view there are only two interesting non-equivalent cases in the theory of μ-representable graphs, namely, those of μ=11 and μ=12
On the neighbour sum distinguishing index of planar graphs

Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers
$1,2,\ldots,k$. Then $k\geq \Delta(G)$, while by Vizing's theorem, no more than
$k=\Delta(G)+1$ is necessary for constructing such $c$. On the course of
investigating irregularities in graphs, it has been moreover conjectured that
only slightly larger $k$, i.e., $k=\Delta(G)+2$ enables enforcing additional
strong feature of $c$, namely that it attributes distinct sums of incident
colours to adjacent vertices in $G$ if only this graph has no isolated edges
and is not isomorphic to $C_5$. We prove the conjecture is valid for planar
graphs of sufficiently large maximum degree. In fact even stronger statement
holds, as the necessary number of colours stemming from the result of Vizing is
proved to be sufficient for this family of graphs. Specifically, our main
result states that every planar graph $G$ of maximum degree at least $28$ which
contains no isolated edges admits a proper edge colouring
$c:E\to\{1,2,\ldots,\Delta(G)+1\}$ such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni
v}c(e)$ for every edge $uv$ of $G$.Comment: 22 page
Bounding the clique-width of H-free chordal graphs.

A graph is H-free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le, and Mosca erroneously claimed that the gem and co-gem are the only two 1-vertex P4-extensions H for which the class of H-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we find four new classes of H-free chordal graphs of bounded clique-width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H-free chordal graphs has bounded clique-width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of inline image-free graphs has bounded clique-width via a reduction to K4-free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique-width of H-free weakly chordal graphs
Flows and bisections in cubic graphs

A $k$-weak bisection of a cubic graph $G$ is a partition of the vertex-set of
$G$ into two parts $V_1$ and $V_2$ of equal size, such that each connected
component of the subgraph of $G$ induced by $V_i$ ($i=1,2$) is a tree of at
most $k-2$ vertices. This notion can be viewed as a relaxed version of
nowhere-zero flows, as it directly follows from old results of Jaeger that
every cubic graph $G$ with a circular nowhere-zero $r$-flow has a $\lfloor r
\rfloor$-weak bisection. In this paper we study problems related to the
existence of $k$-weak bisections. We believe that every cubic graph which has a
perfect matching, other than the Petersen graph, admits a 4-weak bisection and
we present a family of cubic graphs with no perfect matching which do not admit
such a bisection. The main result of this article is that every cubic graph
admits a 5-weak bisection. When restricted to bridgeless graphs, that result
would be a consequence of the assertion of the 5-flow Conjecture and as such it
can be considered a (very small) step toward proving that assertion. However,
the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio
Generation of cubic graphs and snarks with large girth

We describe two new algorithms for the generation of all non-isomorphic cubic
graphs with girth at least $k\ge 5$ which are very efficient for $5\le k \le 7$
and show how these algorithms can be efficiently restricted to generate snarks
with girth at least $k$.
  Our implementation of these algorithms is more than 30, respectively 40 times
faster than the previously fastest generator for cubic graphs with girth at
least 6 and 7, respectively.
  Using these generators we have also generated all non-isomorphic snarks with
girth at least 6 up to 38 vertices and show that there are no snarks with girth
at least 7 up to 42 vertices. We present and analyse the new list of snarks
with girth 6.Comment: 27 pages (including appendix
Graph properties, graph limits, and entropy

We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the coloring number, which by well-known results describes the rate of growth. We study also random graphs and their entropies. We show, for example, that if a hereditary property has a unique limiting graphon with maximal entropy, then a random graph with this property, selected uniformly at random from all such graphs with a given order, converges to this maximizing graphon as the order tends to infinity. © 2017 Wiley Periodicals, Inc