A short proof of the rook reciprocity theorem.
A short WZ-style proof of Abel's identity.
A shorter proof of the distance energy of complete multipartite graphs
Caporossi, Chasser and Furtula in [Les Cahiers du GERAD (2009) G-2009-64] conjectured that the distance energy of a complete multipartite graph of order n with r ≥ 2 parts, each of size at least 2, is equal to 4(n − r). Stevanovic, Milosevic, Hic and Pokorny in [MATCH Commun. Math. Comput. Chem. 70 (2013), no. 1, 157-162.] proved the conjecture, and then Zhang in [Linear Algebra Appl. 450 (2014), 108-120.] gave another proof. We give a shorter proof of this conjecture using the interlacing inequalities...
A simple algorithm for constructing Szemerédi's regularity partition.
A Simple and Relatively Efficient Triangulation of the n-Cube.
A simple bijection between binary trees and colored ternary trees.
A simple card guessing game revisited.
A simple characterization of sets satisfying the central sets theorem.
A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs.
A simple linear algorithm for the connected domination problem in circular-arc graphs
A connected dominating set of a graph G = (V,E) is a subset of vertices CD ⊆ V such that every vertex not in CD is adjacent to at least one vertex in CD, and the subgraph induced by CD is connected. We show that, given an arc family F with endpoints sorted, a minimum-cardinality connected dominating set of the circular-arc graph constructed from F can be computed in O(|F|) time.
A simple method for constructing small cubic graphs of girths 14, 15, and 16.
A simple method for estimating the bounds of spectral radius of nonnegative irreducible matrices.
A simple proof of a theorem by Tutte.
A simple proof of the Aztec diamond theorem.
A Simple Proof of the Perfect Matching Theorem
A simple proof of Whitney's Theorem on connectivity in graphs
In 1932 Whitney showed that a graph with order is 2-connected if and only if any two vertices of are connected by at least two internally-disjoint paths. The above result and its proof have been used in some Graph Theory books, such as in Bondy and Murty’s well-known Graph Theory with Applications. In this note we give a much simple proof of Whitney’s Theorem.
A simple symmetry generating operads related to rooted planar -ary trees and polygonal numbers.
A small trivalent graph of girth 14.
A Sokoban-type game and arc deletion within irregular digraphs of all sizes
Digraphs in which ordered pairs of out- and in-degrees of vertices are mutually distinct are called irregular, see Gargano et al. [3]. Our investigations focus on the problem: what are possible sizes of irregular digraphs (oriented graphs) for a given order n? We show that those sizes in both cases make up integer intervals. The extremal sizes (the endpoints of these intervals) are found in [1,5]. In this paper we construct, with help of Sokoban-type game, n-vertex irregular oriented graphs (irregular...
A solution of Dedekind's problem on the number of isotone Boolean functions.