A note on a distance bound using eigenvalues of the normalized Laplacian matrix.
A note on a Moore bound for graphs embedded in surfaces.
A note on a new condition implying pancyclism
We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to .
A note on a paper by A. Brandt and Y. Intrator
A Note On Analogies Between Characteristic And The Matching Polynomial Of A Graph
A note on another construction of graphs with vertices and cyclic automorphism group of order
The problem of finding minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic groups. This problem was considered earlier by other authors. We give a construction of an undirected graph having vertices and automorphism group cyclic of order , . As a special case we get graphs with vertices and cyclic automorphism groups of order . It can revive interest in related problems.
A note on arc-disjoint cycles in tournaments
We prove that every vertex v of a tournament T belongs to at least arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and (or ) is the out-degree (resp. in-degree) of v.
A Note on Barnette’s Conjecture
Barnette conjectured that each planar, bipartite, cubic, and 3-connected graph is hamiltonian. We prove that this conjecture is equivalent to the statement that there is a constant c > 0 such that each graph G of this class contains a path on at least c|V (G)| vertices.
A note on careful packing of a graph
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 edge-disjoint placement of two copies of G into Kₙ. We prove that with the same condition on size of G we have actually (with few exceptions) a careful packing of G, that is an edge-disjoint placement of two copies of G into Kₙ∖Cₙ.
A note on choosability in planar graphs
A note on chromatic number of direct product of graphs
A note on circuit graphs.
A note on commuting graphs for symmetric groups.
A note on constructing large Cayley graphs of given degree and diameter by voltage assignments.
A Note On Constructions Of Graphs By Means Of Their Spectra
A note on cycle double covers in Cayley graphs.
A note on cyclic chromatic number
A cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that for any 3-connected plane graph G with maximum face degree Δ*. It is known that the conjecture holds true for Δ* ≤ 4 and Δ* ≥ 18. The validity of the conjecture is proved in the paper for some special classes...
A note on degree-continuous graphs
The minimum orders of degree-continuous graphs with prescribed degree sets were investigated by Gimbel and Zhang, Czechoslovak Math. J. 51 (126) (2001), 163–171. The minimum orders were not completely determined in some cases. In this note, the exact values of the minimum orders for these cases are obtained by giving improved upper bounds.
A note on dimension of
A note on divisibility of the number of matchings of a family of graphs.