Independence and matching number in graphs with maximum degree 4
We prove that 74@a(G)+@b(G)>=n(G) and @a(G)+32@b(G)>=n(G) for every triangle-free graph G with maximum degree at most 4, where @a(G) is the independence number and @b(G) is the matching number of G, respectively. These results are sharp for a graph on ...
The isomorphism problem for rose window graphs
Given integers n>=3 and 1@__ __a,r@__ __n-1 with r<>n/2, the rose window graph R"n(a,r) is the graph with vertex set {A"i,B"i|i@__ __{0,1,...,n-1}} and edges {A"i,A"i"+"1},{A"i,B"i},{A"i"+"a,B"i} and {B"i,B"i"+"r} for every i@__ __{0,1,...,n-1} where ...
Simplotopal maps and necklace splitting
We show how to prove combinatorially the Splitting Necklace Theorem by Alon for any number of thieves. Such a proof requires developing a combinatorial theory for abstract simplotopal complexes and simplotopal maps, which generalizes the theory of ...
Second kind maximum matching graph
The second kind maximum matching graphM"2(G) of a graph G is the graph whose vertices are the maximum matchings of G such that two vertices M"1 and M"2 of M"2(G) are adjacent if and only if the symmetric difference of M"1 and M"2 induces either a cycle ...
On 2-factors with a bounded number of odd components
A 2-factor in a graph is a spanning 2-regular subgraph, or equivalently a spanning collection of disjoint cycles. In this paper we investigate the existence of 2-factors with a bounded number of odd cycles in a graph. We extend results of Ryjacek, Saito,...
Another proof of Wilmes' conjecture
We present a new proof of the monomial case of Wilmes' conjecture, which gives a formula for the coarsely-graded Betti numbers of the G-parking function ideal in terms of maximal parking functions of contractions of G. Our proof is via poset topology ...
Counting signed permutations by their alternating runs
The purpose of this paper is to establish a connection between alternating runs of signed permutations in the hyperoctahedral group and left peaks of permutations in the symmetric group, and then to study some associated enumerative polynomials of ...
Two results about Matula numbers
In Matula (1968), D.W. Matula described a bijection between N and the set of rooted trees; the number is called the Matula number of the rooted tree. The Gutman-Ivic-Matula (GIM) function g(n) computes the number of edges of the tree with Matula number ...
On balanced colorings of sparse hypergraphs
We investigate 2-balanced colorings of sparse hypergraphs. As applications, we derive several results on balanced edge-colorings of multigraphs.
A characterization of hypergraphs that achieve equality in the Chvátal-McDiarmid Theorem
For k>=2, let H be a k-uniform hypergraph on n vertices and m edges. The transversal number @t(H) of H is the minimum number of vertices that intersect every edge. Chvatal and McDiarmid (1992) proved that @t(H)@?(n+@?k2@?m)/(@?3k2@?). When k=3, the ...
The Ramsey numbers of wheels versus odd cycles
Given two graphs G"1 and G"2, the Ramsey number R(G"1,G"2) is the smallest integer N such that for any graph G of order N, either G contains G"1 or its complement contains G"2. Let C"m denote a cycle of order m and W"n a wheel of order n+1. In this ...
A note on the saturation number of the family of k-connected graphs
Given a family of graphs F, a graph G is F-saturated if no member of F is a subgraph of G, but for all e@?E(G@?), some member of F is a subgraph of G+e. The saturation number of F, denoted by sat(n,F), is the minimum number of edges in an n-vertex F-...
