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Path Factors and Neighborhoods of Independent Sets in Graphs
A path-factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. Let k ≥ 2 be an integer. A P ≥ k -factor...
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Rainbow Pancyclicity in a Collection of Graphs Under the Dirac-type Condition
Let G = { G i : i ∈ [ n ]} be a collection of not necessarily distinct n -vertex graphs with the same vertex set V , where G can be seen as an edge-colored...
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Saturation numbers for linear forests P6 + tP2
A graph G is H -saturated if it contains no H as a subgraph, but does contain H after the addition of any edge in the complement of G . The saturation...
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A modified similarity degree for C*-algebras
We define variants of Pisier's similarity degree for unital C*-algebras and use direct integral theory to obtain new results. We prove that if every II
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On Gardner’s Conjecture
Gardner conjectured that if two bounded measurable sets A, B ⊂ ℝ n are equidecomposable by a set of isometries Γ generating an amenable group then A ...
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Nonseparating Independent Sets and Maximum Genus of Graphs
A subset I of vertices of an undirected connected graph G is a nonseparating independent set (NSIS) if no two vertices of I are adjacent and G − I is...
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On P≥3-factor Deleted Graphs
A spanning subgraph F of a graph G is called a path factor of G if each component of F is a path. A P ≥ k -factor means a path factor with each...
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Weak Graph Map Homotopy and Its Applications
The authors introduce a notion of a weak graph map homotopy (they call it M -homotopy), discuss its properties and applications. They prove that the...
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Commutativity Preservers of Incidence Algebras
Let I ( X , K ) be the incidence algebra of a finite connected poset X over a field K and D ( X , K ) its subalgebra consisting of diagonal elements. We...
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On Disjoint Cycles of the Same Length in Tournaments
A tournament is an orientation of the complete graph. Tournaments form perhaps the most interesting class of digraphs and it has a great potential...
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Degree sums of adjacent vertices for traceability of claw-free graphs
The line graph of a graph G , denoted by L ( G ), has E ( G ) as its vertex set, where two vertices in L ( G ) are adjacent if and only if the corresponding...
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Some Existence Theorems on Path Factors with Given Properties in Graphs
A path factor of G is a spanning subgraph of G such that its each component is a path. A path factor is called a P ≥ n -factor if its each component...
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Kissing Numbers of Regular Graphs
We prove a sharp upper bound on the number of shortest cycles contained inside any connected graph in terms of its number of vertices, girth, and...
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A Generalization of Implicit Ore-condition for Hamiltonicity of k-connected Graphs
In 2005, Flandrin et al. proved that if G is a k -connected graph of order n and V ( G ) = X 1 ∪ X 2 ∪ ⋯ UXfc such that d ( x ) + d ( y ) ≥ n for each pair of...
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Morse-Bott functions with two critical values on a surface
We study Morse-Bott functions with two critical values (equivalently, non-constant without saddles) on closed surfaces. We show that only four...
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Finding a Shortest Non-Zero Path in Group-Labeled Graphs
We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. ...
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Packing Directed Cycles Quarter- and Half-Integrally
The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback...
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Cycle Lengths in Expanding Graphs
For a positive constant α a graph G on n vertices is called an α-expander if every vertex set U of size at most n /2 has an external neighborhood...
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Melnikov functions in the rigid body dynamics
we review our recent results about perturbations of two cases in the rigid body dynamics: the hess–appelrot case and the lagrange case.