A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r?1 times larger than its matching number. Despite considerable effort, particularly in intersecting case, this conjecture remains wide open, motivating pursuit variants original conjecture. Recently, Bustamante and Stein and, independently, Király Tóthmérész consider...

‎‎we investigate graham higman's paper emph{enumerating }$p$emph{-groups}‎, ‎ii‎, ‎in which he formulated his famous porc conjecture‎. ‎we are able to simplify some of the theory‎. ‎in particular‎, ‎higman's paper contains five pages of homological algebra which he uses in‎ ‎his proof that the number of solutions in a finite field to a finite set of‎ ‎emph{monomial} equations is porc‎. ‎it turn...

We disprove a conjecture of Nagy on the maximum number copies $N$($G$, $H$) fixed graph $G$ in large $H$ with prescribed edge density. conjectured that for all $G$, quantit...

let $gamma(s_n)$ be the minimum number of proper subgroups‎ ‎$h_i, i=1‎, ‎dots‎, ‎l $ of the symmetric group $s_n$ such that each element in $s_n$‎ ‎lies in some conjugate of one of the $h_i.$ in this paper we‎ ‎conjecture that $$gamma(s_n)=frac{n}{2}left(1-frac{1}{p_1}right)‎ ‎left(1-frac{1}{p_2}right)+2,$$ where $p_1,p_2$ are the two smallest primes‎ ‎in the factorization of $ninmathbb{n}$ an...

By Ulam’s conjecture every finite graph G can be reconstructed from its deck of vertex deleted subgraphs. The conjecture is still open, but many special cases have been settled. In particular, one can reconstruct Cartesian products. We consider the case of k-vertex deleted subgraphs of Cartesian products, and prove that one can decide whether a graph H is a kvertex deleted subgraph of a Cartesi...

Two graphs G and H are hypomorphic if there exists a bijection φ : V (G)→ V (H) such that G− v ∼= H − φ(v) for each v ∈ V (G). A graph G is reconstructible if H ∼= G for all H hypomorphic to G. It is well known that not all infinite graphs are reconstructible. However, the Harary-Schwenk-Scott Conjecture from 1972 suggests that all locally finite trees are reconstructible. In this paper, we con...

We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that π(G × H) ≤ π(G)π(H), where π(G) is the pebbling number of the graph G. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that show that Sjöstrand’s theorem on cover p...

Vizing conjectured in 1963 that γ(G2H) > γ(G)γ(H) for any graphs G and H. A graph G is said to satisfy Vizing’s conjecture if the conjectured inequality holds for G and any graph H. Vizing’s conjecture has been proved for γ(G) 6 3, and it is known to hold for other classes of graphs. Clark and Suen in 2000 showed that γ(G2H) > 12γ(G)γ(H) for any graphs G and H. We give a slight improvement of t...

The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger’s Conjecture states that h(G) ≥ χ(G). Since χ(G)α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that α(G)h(G) ≥ |V (G)|. We show that (2α(G) − ⌈log τ (τα(G)/2)⌉)h(G) ≥ |V (G)| where τ ≈ 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of Kawarabayashi and Song who showed (2α(G) − 2)h(G) ≥...

 In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...