A Survey of the Path Partition Conjecture
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paper_title: On a cycle partition problem

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Let G be any graph and let c(G) denote the circumference of G. We conjecture that for every pair c"1,c"2 of positive integers satisfying c"1+c"2=c(G), the vertex set of G admits a partition into two sets V"1 and V"2, such that V"i induces a graph of circumference at most c"i, i=1,2. We establish various results in support of the conjecture; e.g. it is observed that planar graphs, claw-free graphs, certain important classes of perfect graphs, and graphs without too many intersecting long cycles, satisfy the conjecture. This work is inspired by a well-known, long-standing, analogous conjecture involving paths.
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paper_title: Uniquely (m, k)τ-colourable graphs and k − τ-saturated graphs

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Abstract For a graph G , the path number τ ( G ) is defined as the order of a longest path in G . An ( m , k ) τ -colouring of a graph H is a partition of the vertex set of H into m subsets such that each subset induces a subgraph of H for which τ is at most k . The k − τ -chromatic number χ k τ ( H ) is the least m for which H has an ( m , k ) τ -colouring. A graph H is uniquely ( m , k ) τ -colourable if χ k τ ( H ) = m and there is only one partition of the vertex set of H which is an ( m , k ) τ -colouring of H . A graph G is called k − τ -saturated if τ ( G ) ⩽ k and τ ( G + e ) ⩾ k + 1 for all e ϵ E ( G ). For k = 1, the graphs obtained by taking the join of k − τ -saturated graphs (which are empty graphs in this case) are known to be uniquely colourable graphs. In this paper we construct uniquely ( m , k ) τ -colourable graphs (for all positive integers m and k ) using k − τ -saturated graphs in a similar fashion. As a corollary we characterise those p for which there exists a uniquely ( m , k ) τ -colourable graph of order p .
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paper_title: P_{m}-saturated graphs with minimum size

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By \(P_m\) we denote a path of order \(m\). A graph \(G\) is said to be \(P_m\)-saturated if \(G\) has no subgraph isomorphic to \(P_m\) and adding any new edge to \(G\) creates a \(P_m\) in \(G\). In 1986 L. Kaszonyi and Zs. Tuza considered the following problem: for given \(m\) and \(n\) find the minimum size \(sat(n;P_m)\) of \(P_m\)-saturated graph and characterize the graphs of \(Sat(n;P_m)\) - the set of \(P_m\)-saturated graphs of minimum size. They have solved this problem for \(n\geq a_m\) where \(a_m=\begin{cases}3\cdot 2^{k-1}-2 &\quad\text{ if }\quad m=2k,\, k\gt 2\\ 2^{k+1}-2 &\quad\text{ if }\quad m=2k+1,\, k\geq 2\end{cases}\). We define \(b_m=\begin{cases}3\cdot 2^{k-2} &\quad\text{ if }\quad m=2k,\, k\geq 3\\ 3\cdot 2^{k-1}-1 &\quad\text{ if }\quad m=2k+1,\, k\geq 3\end{cases}\) and give \(sat(n;P_m)\) and \(Sat(n;P_m)\) for \(m\geq 6\) and \(b_m\leq n\leq a_m\).
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paper_title: A note on a cycle partition problem

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Abstract Let G be any graph, and let c ( G ) denote the circumference of G . If, for every pair c 1 , c 2 of positive integers satisfying c 1 + c 2 = c ( G ) , the vertex set of G admits a partition into two sets V 1 and V 2 such that V i induces a graph of circumference at most c i , i = 1 , 2 , then G is said to be c -partitionable. In [M.H. Nielsen, On a cycle partition problem, Discrete Math. 308 (2008) 6339–6347], it is conjectured that every graph is c -partitionable. In this paper, we verify this conjecture for a graph with a longest cycle that is a dominating cycle. Moreover, we prove that G is c -partitionable if c ( G ) ≥ | V ( G ) | − 3 .
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paper_title: Saturated graphs with minimal number of edges

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Let F = {F1,…} be a given class of forbidden graphs. A graph G is called F-saturated if no Fi ∈ F is a subgraph of G but the addition of an arbitrary new edge gives a forbidden subgraph. In this paper the minimal number of edges in F-saturated graphs is examined. General estimations are given and the structure of minimal graphs is described for some special forbidden graphs (stars, paths, m pairwise disjoint edges).
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paper_title: On a tree-partition problem

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Abstract If T = ( V , E ) is a tree then – T denotes the additive hereditary property consisting of all graphs that does not contain T as a subgraph. For an arbitrary vertex v of T we deal with a partition of T into two trees T 1 , T 2 , so that V ( T 1 ) ∩ V ( T 2 ) = { v } , V ( T 1 ) ∪ ( T 2 ) = V ( T ) , E ( T 1 ) ∩ E ( T 2 ) = ∅ , E ( T 1 ) ∪ E ( T 2 ) = E ( T ) , T [ V ( T 1 ) \ { v } ] ⊆ E ( T 1 ) and T [ V ( T 2 ) \ { v } ] ⊆ E ( T 2 ) . We call such a partition a T v − p a r t i t i o n of T . We study the following em: Given a graph G belonging to –T . Is it true that for any T v -partition T 1 , T 2 of T there exists a partition { V 1 , V 2 } of the vertices of G such that G [ V 1 ] ∈ − T 1 and G [ V 2 ] ∈ − T 2 ? This problem provides a natural generalization of Δ-partition problem studied by L. Lovasz ([L. Lovasz, On decomposition of graphs. Studia Sci. Math. Hungar. 1 (1966) 237–238]) and Path Partition Conjecture formulated by P. Mihok ([P. Mihok, Problem 4, in: M. Borowiecki, Z. Skupien (Eds.), Graphs, Hypergraphs and Matroids, Zielona Gora, 1985, p. 86]). We present some partial results and a contribution to the Path Kernel Conjecture that was formulated with connection to Path Partition Conjecture.
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paper_title: A note on the Path Kernel Conjecture

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Let @t(G) denote the number of vertices in a longest path in a graph G=(V,E). A subset K of V is called a P"n-kernel of G if @t(G[K])@?n-1 and every vertex [email protected][email protected]?K is adjacent to an end-vertex of a path of order n-1 in G[K]. It is known that every graph has a P"n-kernel for every positive integer [email protected]?9. R. Aldred and C. Thomassen in [R.E.L. Aldred, C. Thomassen, Graphs with not all possible path-kernels, Discrete Math. 285 (2004) 297-300] proved that there exists a graph which contains no P"3"6"4-kernel. In this paper, we generalise this result. We construct a graph with no P"1"5"5-kernel and for each integer l>=0 we provide a construction of a graph G containing no P"@t"("G")"-"l-kernel.
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paper_title: Graphs with not all possible path-kernels

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Abstract The Path Partition Conjecture states that the vertices of a graph G with longest path of length c may be partitioned into two parts X and Y such that the longest path in the subgraph of G induced by X has length at most a and the longest path in the subgraph of G induced by Y has length at most b , where a + b = c . Moreover, for each pair a , b such that a + b = c there is a partition with this property. A stronger conjecture by Broere, Hajnal and Mihok, raised as a problem by Mihok in 1985, states the following: For every graph G and each integer k , c ⩾ k ⩾2 there is a partition of V ( G ) into two parts (K, K ) such that the subgraph G [ K ] of G induced by K has no path on more than k −1 vertices and each vertex in K is adjacent to an endvertex of a path on k −1 vertices in G [ K ]. In this paper we provide a counterexample to this conjecture.
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paper_title: Graph coloring problems

paper_content:

Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.
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paper_title: A survey of hereditary properties of graphs

paper_content:

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.
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paper_title: Cycles in k-traceable oriented graphs

paper_content:

A digraph of order at least k is termed k-traceable if each of its subdigraphs of order k is traceable. It turns out that several properties of tournaments-i.e., the 2-traceable oriented graphs-extend to k-traceable oriented graphs for small values of k. For instance, the authors together with O. Oellermann have recently shown that for k=2,3,4,5,6, all k-traceable oriented graphs are traceable. Moon [J.W. Moon, On subtournaments of a tournament, Canad. Math. Bull. 9(3) (1966) 297-301] observed that every nontrivial strong tournament T is vertex-pancyclic-i.e., through each vertex there is a cycle of every length from 3 up to the order of T. The present paper reports results pertaining to various cycle properties of strong k-traceable oriented graphs and explores the extent to which pancyclicity is retained by strong k-traceable oriented graphs. For each k>=2 there are infinitely many k-traceable oriented graphs-e.g. tournaments. However, we establish an upper bound (linear in k) on the order of k-traceable oriented graphs having a strong component with girth greater than 3. As an application of our findings, we show that the Path Partition Conjecture holds for 1-deficient oriented graphs having a strong component with girth at least 6. (A digraph is 1-deficient if its order is exactly one more than the order of its longest paths.)
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paper_title: A note on path kernels and partitions

paper_content:

The detour order of a graph G, denoted by @t(G), is the order of a longest path in G. A subset S of V(G) is called a P"n-kernel of G if @t(G[S])@?n-1 and every vertex v@?V(G)-S is adjacent to an end-vertex of a path of order n-1 in G[S]. A partition of the vertex set of G into two sets, A and B, such that @t(G[A])@?a and @t(G[B])@?b is called an (a,b)-partition of G. In this paper we show that any graph with girth g has a P"n"+"1-kernel for every n<3g2-1. Furthermore, if @t(G)=a+b, 1@?a@?b, and G has girth greater than 23(a+1), then G has an (a,b)-partition.
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paper_title: Longest path partitions in generalizations of tournaments

paper_content:

We consider the so-called Path Partition Conjecture for digraphs which states that for every digraph, D, and every choice of positive integers, @l"1,@l"2, such that @l"1+@l"2 equals the order of a longest directed path in D, there exists a partition of D into two digraphs, D"1 and D"2, such that the order of a longest path in D"i is at most @l"i, for i=1,2. We prove that certain classes of digraphs, which are generalizations of tournaments, satisfy the Path Partition Conjecture and that some of the classes even satisfy the conjecture with equality.
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paper_title: Stable set meeting every longest path

paper_content:

Laborde, Payan and Xuong conjectured that every digraph has a stable set meeting every longest path. We prove that this conjecture holds for digraphs with stability number at most 2.
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paper_title: A Traceability Conjecture for Oriented Graphs

paper_content:

A (di)graph $G$ of order $n$ is $k$-traceable (for some $k$, $1\leq k\leq n$) if every induced sub(di)graph of $G$ of order $k$ is traceable. It follows from Dirac's degree condition for hamiltonicity that for $k\geq2$ every $k$-traceable graph of order at least $2k-1$ is hamiltonian. The same is true for strong oriented graphs when $k=2,3,4,$ but not when $k\geq5$. However, we conjecture that for $k\geq2$ every $k$-traceable oriented graph of order at least $2k-1$ is traceable. The truth of this conjecture would imply the truth of an important special case of the Path Partition Conjecture for Oriented Graphs. In this paper we show the conjecture is true for $k \leq 5$ and for certain classes of graphs. In addition we show that every strong $k$-traceable oriented graph of order at least $6k-20$ is traceable. We also characterize those graphs for which all walkable orientations are $k$-traceable.
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paper_title: Progress on the Traceability Conjecture for Oriented Graphs

paper_content:

A digraph is k -traceable if each of its induced subdigraphs of order k is traceable. The Traceability Conjecture is that for k ≥ 2 every k -traceable oriented graph of order at least 2k-1 is traceable. The conjecture has been proved for k ≤ 5 . We prove that it also holds for k=6 .
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paper_title: Combinatorics and graph theory

paper_content:

Graph Theory: Introductory Concepts.- Trees.- Planarity.- Colorings.- Matchings.- Ramsey Theory.- References Combinatorics: Three Basic Problems.- Binomial Coefficients.- The Principle of Inclusion and Exclusion.- Generating Functions.- Polya's Theory of Counting.- More Numbers.- Stable Marriage.- References Infinite Combinatorics and Graph Theory: Pigeons and Trees.- Ramsey Revisited.- ZFC.- The Return of der Koenig.- Ordinals, Cardinals, and Many Pigeons.- Incompleteness and Coardinals.- Weakly Compact Cardinals.- Finite Combinatorics with Infinite Consequences.- Points of Departure.- References.
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paper_title: The Path Partition Conjecture is true for some generalizations of tournaments

paper_content:

Abstract The Path Partition Conjecture for digraphs states that for every digraph D , and every choice of positive integers λ 1 , λ 2 such that λ 1 + λ 2 equals the order of a longest directed path in D , there exists a partition of D in two subdigraphs D 1 , D 2 such that the order of the longest path in D i is at most λ i for i = 1 , 2 . We present sufficient conditions for a digraph to satisfy the Path Partition Conjecture. Using these results, we prove that strong path mergeable, arc-locally semicomplete, strong 3-quasi-transitive, strong arc-locally in-semicomplete and strong arc-locally out-semicomplete digraphs satisfy the Path Partition Conjecture. Some previous results are generalized.
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paper_title: Traceability of k-traceable oriented graphs

paper_content:

A digraph of order at least k is k-traceable if each of its subdigraphs of order k is traceable. We note that 2-traceable oriented graphs are tournaments and for k>=3, k-traceable oriented graphs can be regarded as generalized tournaments. We show that for [email protected][email protected]?6 every k-traceable oriented graph is traceable, thus extending the well-known fact that every tournament is traceable. This result does not extend to k=7. In fact, for every k>=7, except possibly for k=8 or 10, there exist k-traceable oriented graphs that are nontraceable. However, we show that for every k>=2 there exists a smallest integer t(k) such that every k-traceable oriented graph of order at least t(k) is traceable.
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paper_title: Digraphs -- Theory, Algorithms and Applications

paper_content:

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.
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paper_title: The Path Partition Conjecture is true for claw-free graphs

paper_content:

The detour order of a graph G, denoted by @t(G), is the order of a longest path in G. The Path Partition Conjecture (PPC) is the following: If G is any graph and (a,b) any pair of positive integers such that @t(G)=a+b, then the vertex set of G has a partition (A,B) such that @t()=)=
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paper_title: A survey of hereditary properties of graphs

paper_content:

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.
---
paper_title: A note on path kernels and partitions

paper_content:

The detour order of a graph G, denoted by @t(G), is the order of a longest path in G. A subset S of V(G) is called a P"n-kernel of G if @t(G[S])@?n-1 and every vertex v@?V(G)-S is adjacent to an end-vertex of a path of order n-1 in G[S]. A partition of the vertex set of G into two sets, A and B, such that @t(G[A])@?a and @t(G[B])@?b is called an (a,b)-partition of G. In this paper we show that any graph with girth g has a P"n"+"1-kernel for every n<3g2-1. Furthermore, if @t(G)=a+b, 1@?a@?b, and G has girth greater than 23(a+1), then G has an (a,b)-partition.
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paper_title: A note on path kernels and partitions

paper_content:

The detour order of a graph G, denoted by @t(G), is the order of a longest path in G. A subset S of V(G) is called a P"n-kernel of G if @t(G[S])@?n-1 and every vertex v@?V(G)-S is adjacent to an end-vertex of a path of order n-1 in G[S]. A partition of the vertex set of G into two sets, A and B, such that @t(G[A])@?a and @t(G[B])@?b is called an (a,b)-partition of G. In this paper we show that any graph with girth g has a P"n"+"1-kernel for every n<3g2-1. Furthermore, if @t(G)=a+b, 1@?a@?b, and G has girth greater than 23(a+1), then G has an (a,b)-partition.
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paper_title: The Path Partition Conjecture is true for claw-free graphs

paper_content:

The detour order of a graph G, denoted by @t(G), is the order of a longest path in G. The Path Partition Conjecture (PPC) is the following: If G is any graph and (a,b) any pair of positive integers such that @t(G)=a+b, then the vertex set of G has a partition (A,B) such that @t()=)=
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paper_title: On a Closure Concept in Claw-Free Graphs

paper_content:

IfGis a claw-free graph, then there is a graphcl(G) such that (i)Gis a spanning subgraph ofcl(G), (ii)cl(G) is a line graph of a triangle-free graph, and (iii)the length of a longest cycle inGand incl(G) is the same. A sufficient condition for hamiltonicity in claw-free graphs, the equivalence of some conjectures on hamiltonicity in 2-tough graphs and the hamiltonicity of 7-connected claw-free graphs are obtained as corollaries.
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paper_title: Closure and stable Hamiltonian properties in claw-free graphs

paper_content:

In the class of k-connected claw-free graphs, we study the stability of some Hamiltonian properties under a closure operation introduced by the third author. We prove that (i) the properties of pancyclicity, vertex pancyclicity and cycle extendability are not stable for any k (i.e., for any of these properties there is an infinite family of graphs Gk of arbitrarily high connectivity k such that the closure of Gk has the property while the graph Gk does not); (ii) traceability is a stable property even for k = 1; (iii) homogeneous traceability is not stable for k = 2 (although it is stable for k = 7). The article is concluded with several open questions concerning stability of homogeneous traceability and Hamiltonian connectedness. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 30–41, 2000
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paper_title: The Path Partition Conjecture is true for claw-free graphs

paper_content:

The detour order of a graph G, denoted by @t(G), is the order of a longest path in G. The Path Partition Conjecture (PPC) is the following: If G is any graph and (a,b) any pair of positive integers such that @t(G)=a+b, then the vertex set of G has a partition (A,B) such that @t()=)=
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paper_title: Cycles in k-traceable oriented graphs

paper_content:

A digraph of order at least k is termed k-traceable if each of its subdigraphs of order k is traceable. It turns out that several properties of tournaments-i.e., the 2-traceable oriented graphs-extend to k-traceable oriented graphs for small values of k. For instance, the authors together with O. Oellermann have recently shown that for k=2,3,4,5,6, all k-traceable oriented graphs are traceable. Moon [J.W. Moon, On subtournaments of a tournament, Canad. Math. Bull. 9(3) (1966) 297-301] observed that every nontrivial strong tournament T is vertex-pancyclic-i.e., through each vertex there is a cycle of every length from 3 up to the order of T. The present paper reports results pertaining to various cycle properties of strong k-traceable oriented graphs and explores the extent to which pancyclicity is retained by strong k-traceable oriented graphs. For each k>=2 there are infinitely many k-traceable oriented graphs-e.g. tournaments. However, we establish an upper bound (linear in k) on the order of k-traceable oriented graphs having a strong component with girth greater than 3. As an application of our findings, we show that the Path Partition Conjecture holds for 1-deficient oriented graphs having a strong component with girth at least 6. (A digraph is 1-deficient if its order is exactly one more than the order of its longest paths.)
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paper_title: Longest path partitions in generalizations of tournaments

paper_content:

We consider the so-called Path Partition Conjecture for digraphs which states that for every digraph, D, and every choice of positive integers, @l"1,@l"2, such that @l"1+@l"2 equals the order of a longest directed path in D, there exists a partition of D into two digraphs, D"1 and D"2, such that the order of a longest path in D"i is at most @l"i, for i=1,2. We prove that certain classes of digraphs, which are generalizations of tournaments, satisfy the Path Partition Conjecture and that some of the classes even satisfy the conjecture with equality.
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paper_title: Stable set meeting every longest path

paper_content:

Laborde, Payan and Xuong conjectured that every digraph has a stable set meeting every longest path. We prove that this conjecture holds for digraphs with stability number at most 2.
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paper_title: A Traceability Conjecture for Oriented Graphs

paper_content:

A (di)graph $G$ of order $n$ is $k$-traceable (for some $k$, $1\leq k\leq n$) if every induced sub(di)graph of $G$ of order $k$ is traceable. It follows from Dirac's degree condition for hamiltonicity that for $k\geq2$ every $k$-traceable graph of order at least $2k-1$ is hamiltonian. The same is true for strong oriented graphs when $k=2,3,4,$ but not when $k\geq5$. However, we conjecture that for $k\geq2$ every $k$-traceable oriented graph of order at least $2k-1$ is traceable. The truth of this conjecture would imply the truth of an important special case of the Path Partition Conjecture for Oriented Graphs. In this paper we show the conjecture is true for $k \leq 5$ and for certain classes of graphs. In addition we show that every strong $k$-traceable oriented graph of order at least $6k-20$ is traceable. We also characterize those graphs for which all walkable orientations are $k$-traceable.
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paper_title: Progress on the Traceability Conjecture for Oriented Graphs

paper_content:

A digraph is k -traceable if each of its induced subdigraphs of order k is traceable. The Traceability Conjecture is that for k ≥ 2 every k -traceable oriented graph of order at least 2k-1 is traceable. The conjecture has been proved for k ≤ 5 . We prove that it also holds for k=6 .
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paper_title: The Path Partition Conjecture is true for some generalizations of tournaments

paper_content:

Abstract The Path Partition Conjecture for digraphs states that for every digraph D , and every choice of positive integers λ 1 , λ 2 such that λ 1 + λ 2 equals the order of a longest directed path in D , there exists a partition of D in two subdigraphs D 1 , D 2 such that the order of the longest path in D i is at most λ i for i = 1 , 2 . We present sufficient conditions for a digraph to satisfy the Path Partition Conjecture. Using these results, we prove that strong path mergeable, arc-locally semicomplete, strong 3-quasi-transitive, strong arc-locally in-semicomplete and strong arc-locally out-semicomplete digraphs satisfy the Path Partition Conjecture. Some previous results are generalized.
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paper_title: Traceability of k-traceable oriented graphs

paper_content:

A digraph of order at least k is k-traceable if each of its subdigraphs of order k is traceable. We note that 2-traceable oriented graphs are tournaments and for k>=3, k-traceable oriented graphs can be regarded as generalized tournaments. We show that for [email protected][email protected]?6 every k-traceable oriented graph is traceable, thus extending the well-known fact that every tournament is traceable. This result does not extend to k=7. In fact, for every k>=7, except possibly for k=8 or 10, there exist k-traceable oriented graphs that are nontraceable. However, we show that for every k>=2 there exists a smallest integer t(k) such that every k-traceable oriented graph of order at least t(k) is traceable.
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Title: A Survey of the Path Partition Conjecture

Section 1: Introduction
Description 1: Introduce the Path Partition Conjecture (PPC), its significance, history, and main objectives of the survey.

Section 2: The PPC and the Lattice of Additive Hereditary Properties
Description 2: Define key notations and terminologies used in the study, and discuss the relevance of additive hereditary properties to the PPC.

Section 3: W_p-maximal Sets and P_p+2-kernels
Description 3: Explore the concept of W_p-maximal sets and P_p+2-kernels, including related conjectures and theorems.

Section 4: The PPC and Cycle Structure
Description 4: Present the relationship between cycle structures in graphs and the PPC, including important results and corollaries.

Section 5: The PPC Restricted to Special Classes of Graphs
Description 5: Discuss the PPC’s validity for special classes of graphs such as claw-free graphs and planar graphs, including theorems and significant results.

Section 6: The PPC for t-deficient Graphs
Description 6: Investigate the PPC’s implications for t-deficient graphs, providing key theorems and asymptotic results.

Section 7: Generalized Chromatic Numbers
Description 7: Examine the connection between the PPC and generalized chromatic numbers, as well as related conjectures and bounds.

Section 8: Directed and Oriented Analogues of the PPC
Description 8: Briefly discuss the directed and oriented versions of the PPC and their challenges and results achieved thus far.

Section 9: Conclusion
Description 9: Summarize the main findings of the survey, the current state of the PPC, and potential directions for future research.