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1.
Given a nonnegative integer and a positive integer , a graph is said to be -colorable if the vertices of can be colored with colors such that every vertex has at most neighbors receiving the same color as itself. Let be the family of planar graphs without -cycles adjacent to cycles of length 3 or 5. This paper proves that everyone in is -colorable. This is the best possible in the sense that there are members in which are not -colorable. 相似文献
2.
Jeong-Hyun Kang 《Discrete Mathematics》2018,341(1):96-103
The vertices of Kneser graph are the subsets of of cardinality , two vertices are adjacent if and only if they are disjoint. The square of a graph is defined on the vertex set of with two vertices adjacent if their distance in is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when . It is believed that where is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: for 1 (Kim and Park, 2014) and for (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to , where is a constant in , depending on . 相似文献
3.
In this paper, we employed lattice model to describe the three internally vertex-disjoint paths that span the vertex set of the generalized Petersen graph . We showed that the is 3-spanning connected for odd . Based on the lattice model, five amalgamated and one extension mechanisms are introduced to recursively establish the 3-spanning connectivity of the . In each amalgamated mechanism, a particular lattice trail was amalgamated with the lattice trails that was dismembered, transferred, or extended from parts of the lattice trails for , where a lattice tail is a trail in the lattice model that represents a path in . 相似文献
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Let and be two positive integers such that and . A graph is an -parity factor of a graph if is a spanning subgraph of and for all vertices , and . In this paper we prove that every connected graph with vertices has an -parity factor if is even, , and for any two nonadjacent vertices , . This extends an earlier result of Nishimura (1992) and strengthens a result of Cai and Li (1998). 相似文献
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7.
Aysel Erey 《Discrete Mathematics》2018,341(5):1419-1431
8.
David Gilat Isaac Meilijson Laura Sacerdote 《Stochastic Processes and their Applications》2018,128(6):1849-1856
For a martingale starting at with final variance , and an interval , let be the normalized length of the interval and let be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of by is at most if and at most otherwise. Both bounds are sharp, attained by Standard Brownian Motion stopped at appropriate stopping times. Both bounds also attain the Doob upper bound on the expected number of upcrossings of for submartingales with the corresponding final distribution. Each of these two bounds is at most , with equality in the first bound for . The upper bound on the length covered by during upcrossings of an interval restricts the possible variability of a martingale in terms of its final variance. This is in the same spirit as the Dubins & Schwarz sharp upper bound on the expected maximum of above , the Dubins & Schwarz sharp upper bound on the expected maximal distance of from , and the Dubins, Gilat & Meilijson sharp upper bound on the expected diameter of . 相似文献
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Houssein El Turkey 《Journal of Pure and Applied Algebra》2018,222(1):181-190
The complexity of a module is the rate of growth of the minimal projective resolution of the module while the z-complexity is the rate of growth of the number of indecomposable summands at each step in the resolution. Let () be the type II orthosymplectic Lie superalgebra of types B or D. In this paper, we compute the complexity and the z-complexity of the simple finite-dimensional -supermodules. We then give these complexities certain geometric interpretations using support and associated varieties. 相似文献
11.
Johnson proved that if are coprime integers, then the th moment of the size of an -core is a polynomial of degree in for fixed . After that, by defining a statistic size on elements of affine Weyl group, which is preserved under the bijection between minimal coset representatives of and -cores, Thiel and Williams obtained the variance and the third moment about the mean of the size of an -core. Later, Ekhad and Zeilberger stated the first six moments about the mean of the size of an -core and the first nine moments about the mean of the size of an -core using Maple. To get the moments about the mean of the size of a self-conjugate -core, we proceed to follow the approach of Thiel and Williams, however, their approach does not seem to directly apply to the self-conjugate case. In this paper, following Johnson’s approach, by Ehrhart theory and Euler–Maclaurin theory, we prove that if are coprime integers, then the th moment about the mean of the size of a self-conjugate -core is a quasipolynomial of period 2 and degree in for fixed odd . Then, based on a bijection of Ford, Mai and Sze between self-conjugate -cores and lattice paths in rectangle and a formula of Chen, Huang and Wang on the size of self-conjugate -cores, we obtain the variance, the third moment and the fourth moment about the mean of the size of a self-conjugate -core. 相似文献
12.
A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. A -matching in a 3-uniform hypergraph is a matching of size . Let be a partition of vertices such that and . Denote by the 3-uniform hypergraph with vertex set consisting of all those edges which contain at least two vertices of . Let be a 3-uniform hypergraph of order such that for any two adjacent vertices . In this paper, we prove contains a -matching if and only if is not a subgraph of . 相似文献
13.
For integers , a -coloring of a graph is a proper coloring with at most colors such that for any vertex with degree , there are at least min different colors present at the neighborhood of . The -hued chromatic number of , , is the least integer such that a -coloring of exists. The list-hued chromatic number of is similarly defined. Thus if , then . We present examples to show that, for any sufficiently large integer , there exist graphs with maximum average degree less than 3 that cannot be -colored. We prove that, for any fraction , there exists an integer such that for each , every graph with maximum average degree is list -colorable. We present examples to show that for some there exist graphs with maximum average degree less than 4 that cannot be -hued colored with less than colors. We prove that, for any sufficiently small real number , there exists an integer such that every graph with maximum average degree satisfies . These results extend former results in Bonamy et al. (2014). 相似文献
14.
For a subgraph of , let be the maximum number of vertices of that are pairwise distance at least three in . In this paper, we prove three theorems. Let be a positive integer, and let be a subgraph of an -connected claw-free graph . We prove that if , then either can be covered by a cycle in , or there exists a cycle in such that . This result generalizes the result of Broersma and Lu that has a cycle covering all the vertices of if . We also prove that if , then either can be covered by a path in , or there exists a path in such that . By using the second result, we prove the third result. For a tree , a vertex of with degree one is called a leaf of . For an integer , a tree which has at most leaves is called a -ended tree. We prove that if , then has a -ended tree covering all the vertices of . This result gives a positive answer to the conjecture proposed by Kano et al. (2012). 相似文献
15.
Ju Zhou 《Discrete Mathematics》2018,341(4):1021-1031
A graph is induced matching extendable or IM-extendable if every induced matching of is contained in a perfect matching of . In 1998, Yuan proved that a connected IM-extendable graph on vertices has at least edges, and that the only IM-extendable graph with vertices and edges is , where is an arbitrary tree on vertices. In 2005, Zhou and Yuan proved that the only IM-extendable graph with vertices and edges is , where is an arbitrary tree on vertices and is an edge connecting two vertices that lie in different copies of and have distance 3 between them in . In this paper, we introduced the definition of -joint graph and characterized the connected IM-extendable graphs with vertices and edges. 相似文献
16.
Greg Malen 《Discrete Mathematics》2018,341(9):2567-2574
For any fixed graph , we prove that the topological connectivity of the graph homomorphism complex Hom() is at least , where , for the minimum degree of a vertex in a subgraph . This generalizes a theorem of C?uki? and Kozlov, in which the maximum degree was used in place of , and provides a high-dimensional analogue of the graph theoretic bound for chromatic number, , as . Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom when for a fixed constant . 相似文献
17.
The purpose of this note is to show a new series of examples of homogeneous ideals I in for which the containment fails. These ideals are supported on certain arrangements of lines in , which resemble Fermat configurations of points in , see [14]. All examples exhibiting the failure of the containment constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own. 相似文献
18.
In this paper, we consider -cycle decomposition of
and directed -cycle decompositions of and , where and denote the wreath product and tensor product of graphs, respectively. Using the results obtained here, we prove that for , the obvious necessary conditions for the existence of a -decomposition of are sufficient whenever where is a prime and . Also, we show that the necessary conditions for the existence of -decompositions of and are sufficient whenever is a prime, where denotes the directed cycle of length . 相似文献
19.
Let be a prime power and be a positive integer. A subspace partition of , the vector space of dimension over , is a collection of subspaces of such that each nonzero vector of is contained in exactly one subspace in ; the multiset of dimensions of subspaces in is then called a Gaussian partition of . We say that contains a direct sum if there exist subspaces such that . In this paper, we study the problem of classifying the subspace partitions that contain a direct sum. In particular, given integers and with , our main theorem shows that if is a subspace partition of with subspaces of dimension for , then contains a direct sum when has a solution for some integers and belongs to the union of two natural intervals. The lower bound of captures all subspace partitions with dimensions in that are currently known to exist. Moreover, we show the existence of infinite classes of subspace partitions without a direct sum when or when the condition on the existence of a nonnegative integral solution is not satisfied. We further conjecture that this theorem can be extended to any number of distinct dimensions, where the number of subspaces in each dimension has appropriate bounds. These results offer further evidence of the natural combinatorial relationship between Gaussian and integer partitions (when ) as well as subspace and set partitions. 相似文献
20.
A graph is minimally -tough if the toughness of is and the deletion of any edge from decreases the toughness. Kriesell conjectured that for every minimally -tough graph the minimum degree . We show that in every minimally -tough graph . We also prove that every minimally -tough, claw-free graph is a cycle. On the other hand, we show that for every positive rational number any graph can be embedded as an induced subgraph into a minimally -tough graph. 相似文献