In this paper, we study extremal values for the determinant of the weighted graph Laplacian under simple nondegeneracy conditions on the weights. We derive necessary and sufficient conditions for the determinant of the Laplacian to be bounded away from zero and for the existence of a minimizing set of weights. These conditions are given both in terms of properties of random spanning trees and in terms of a type of density on graphs. These results generalize and extend the work of Melikyan.
A graph drawn on the plane is called 1-plane if each edge is crossed at most once by another edge. In this paper, we show that every 4-edge-connected 1-plane graph has a connected spanning plane subgraph. We also show that there exist infinitely many 4-connected 1-plane graphs that have no 2-connected spanning plane subgraphs. Moreover, we consider the condition of and such that every -connected 1-plane graph has an -connected spanning plane subgraph.
For a multiset of positive integers, let . A -list assignment of is a list assignment
For a finite set of integers such that the first few gaps between its consecutive elements equal , while the remaining gaps equal , we study dense packings of its translates on the line. We obtain an explicit lower bound on the corresponding optimal density, conjecture its tightness, and prove it in case one of the gap lengths, or , appears only once. This is equivalent to a Motzkin problem on the independence ratio of certain integer distance graphs.
We show that every commutative idempotent monoid (a.k.a. lattice) is the endomorphism monoid of a subcubic graph. This solves a problem of Babai and Pultr and the degree bound is best-possible. On the other hand, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids. As a by-product, we prove that monoids can be represented by graphs of bounded expansion (reproving a result of Nešetřil and Ossona de Mendez) and -cancellative monoids can be represented by graphs of bounded degree. Finally, we show that not all completely regular monoids can be represented by graphs excluding topological minor (strengthening a result of Babai and Pultr).
Barnette's conjecture asserts that every cubic 3-connected plane bipartite graph is hamiltonian. Although, in general, the problem is still open, some partial results are known. In particular, let us call a face of a plane graph big (small) if it has at least six edges (it has four edges, respectively). Goodey proved for a 3-connected bipartite cubic plane graph , that if all big faces in have exactly six edges, then is hamiltonian. In this paper, we prove that the same is true under the condition that no face in has more than four big neighbours. We also prove, that if each vertex in is incident both with a small and a big face, then has at least different Hamilton cycles, where
Let be a graph in which each edge is assigned one of the colours , and let be a subgroup of . The operation of switching at a vertex of with respect to an element of permutes the colours of the edges incident with according to
A cycle is said to be directed if all its arcs have the same direction. Otherwise, it is said to be nondirected. A strong tournament is a tournament containing a directed path from any vertex to any other vertex. A tournament that is not strong is said to be reducible. Rosenfeld conjectured that there exists an integer such that every tournament of order contains any Hamiltonian nondirected cycle. Havet proved this conjecture for and for reducible tournaments for . Finding non-Hamiltonian cycles seems more simple. Thomason proved that any tournament of order contains any nondirected cycle of order . This implies the existence of cycles of order
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