A note on domino shuffling.
A note on minimally 3-connected graphs
If G is a minimally 3-connected graph and C is a double cover of the set of edges of G by irreducible walks, then |E(G)| ≥ 2| C| - 2.
A Note on Non-Dominating Set Partitions in Graphs
A set S of vertices of a graph G is a dominating set if every vertex not in S is adjacent to a vertex of S and is a total dominating set if every vertex of G is adjacent to a vertex of S. The cardinality of a minimum dominating (total dominating) set of G is called the domination (total domination) number. A set that does not dominate (totally dominate) G is called a non-dominating (non-total dominating) set of G. A partition of the vertices of G into non-dominating (non-total dominating) sets is...
A note on packing graphs without cycles of length up to five.
A note on packing of two copies of a hypergraph
A 2-packing of a hypergraph 𝓗 is a permutation σ on V(𝓗) such that if an edge e belongs to 𝓔(𝓗), then σ (e) does not belong to 𝓔(𝓗). We prove that a hypergraph which does not contain neither empty edge ∅ nor complete edge V(𝓗) and has at most 1/2n edges is 2-packable. A 1-uniform hypergraph of order n with more than 1/2n edges shows that this result cannot be improved by increasing the size of 𝓗.
A note on perfect matchings in uniform hypergraphs with large minimum collective degree
For an integer and a -uniform hypergraph , let be the largest integer such that every -element set of vertices of belongs to at least edges of . Further, let be the smallest integer such that every -uniform hypergraph on vertices and with contains a perfect matching. The parameter has been completely determined for all and large divisible by by Rödl, Ruci’nski, and Szemerédi in [Perfect matchings in large uniform hypergraphs with large minimum collective degree, submitted]....
A note on pm-compact bipartite graphs
A graph is called perfect matching compact (briefly, PM-compact), if its perfect matching graph is complete. Matching-covered PM-compact bipartite graphs have been characterized. In this paper, we show that any PM-compact bipartite graph G with δ (G) ≥ 2 has an ear decomposition such that each graph in the decomposition sequence is also PM-compact, which implies that G is matching-covered
A note on polynomials and -factors of graphs.
A note on the hardness results for the labeled perfect matching problems in bipartite graphs
In this note, we strengthen the inapproximation bound of O(logn) for the labeled perfect matching problem established in J. Monnot, The Labeled perfect matching in bipartite graphs, Information Processing Letters96 (2005) 81–88, using a self improving operation in some hard instances. It is interesting to note that this self improving operation does not work for all instances. Moreover, based on this approach we deduce that the problem does not admit constant approximation algorithms for connected...
A note on the open packing number in graphs
A subset of vertices in a graph is an open packing set if no pair of vertices of has a common neighbor in . An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The maximum cardinality of an open packing set is called the open packing number and is denoted by . A subset in a graph with no isolated vertex is called a total dominating set if any vertex of is adjacent to some vertex of . The total domination number of , denoted...
A Note on the Uniqueness of Stable Marriage Matching
In this note we present some sufficient conditions for the uniqueness of a stable matching in the Gale-Shapley marriage classical model of even size. We also state the result on the existence of exactly two stable matchings in the marriage problem of odd size with the same conditions.
A Note on Uniquely Embeddable Forests
Let F be a forest of order n. It is well known that if F 6= Sn, a star of order n, then there exists an embedding of F into its complement F. In this note we consider a problem concerning the uniqueness of such an embedding.
A note on uniquely embeddable graphs
Let G be a simple graph of order n and size e(G). It is well known that if e(G) ≤ n-2, then there is an embedding G into its complement [G̅]. In this note, we consider a problem concerning the uniqueness of such an embedding.
A path(ological) partition problem
Let τ(G) denote the number of vertices in a longest path of the graph G and let k₁ and k₂ be positive integers such that τ(G) = k₁ + k₂. The question at hand is whether the vertex set V(G) can be partitioned into two subsets V₁ and V₂ such that τ(G[V₁] ) ≤ k₁ and τ(G[V₂] ) ≤ k₂. We show that several classes of graphs have this partition property.
A prime factor theorem for a generalized direct product
We introduce the concept of neighborhood systems as a generalization of directed, reflexive graphs and show that the prime factorization of neighborhood systems with respect to the the direct product is unique under the condition that they satisfy an appropriate notion of thinness.
A proof of the two-path conjecture.
A Property Of The Number Of Perfect Matchings Of A Graph
A ramsey-type theorem for multiple disjoint copies of induced subgraphs
Let k and ℓ be positive integers with ℓ ≤ k − 2. It is proved that there exists a positive integer c depending on k and ℓ such that every graph of order (2k−1−ℓ/k)n+c contains n vertex disjoint induced subgraphs, where these subgraphs are isomorphic to each other and they are isomorphic to one of four graphs: (1) a clique of order k, (2) an independent set of order k, (3) the join of a clique of order ℓ and an independent set of order k − ℓ, or (4) the union of an independent set of order ℓ and...
A remark about fractional -critical graphs.
A result on -critical graphs.