On generating series of coloured planar trees.
On generating sets of induced-hereditary properties
A natural generalization of the fundamental graph vertex-colouring problem leads to the class of problems known as generalized or improper colourings. These problems can be very well described in the language of reducible (induced) hereditary properties of graphs. It turned out that a very useful tool for the unique determination of these properties are generating sets. In this paper we focus on the structure of specific generating sets which provide the base for the proof of The Unique Factorization...
On generating snarks
We discuss the construction of snarks (that is, cyclically 4-edge connected cubic graphs of girth at least five which are not 3-edge colourable) by using what we call colourable snark units and a welding process.
On Genocchi numbers and polynomials.
On Golomb birulers and their applications
On good characterizations for digraphs
On graceful colorings of trees
A proper coloring , of a graph is called a graceful -coloring if the induced edge coloring defined by for each edge of is also proper. The minimum integer for which has a graceful -coloring is the graceful chromatic number . It is known that if is a tree with maximum degree , then and this bound is best possible. It is shown for each integer that there is an infinite class of trees with maximum degree such that . In particular, we investigate for each integer a...
On graceful trees.
On graph associated to co-ideals of commutative semirings
Let be a commutative semiring with non-zero identity. In this paper, we introduce and study the graph whose vertices are all elements of and two distinct vertices and are adjacent if and only if the product of the co-ideals generated by and is . Also, we study the interplay between the graph-theoretic properties of this graph and some algebraic properties of semirings. Finally, we present some relationships between the zero-divisor graph and .
On Graph-Based Cryptography and Symbolic Computations
We have been investigating the cryptographical properties of in nite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arbitrarily large degree are known. The paper is devoted to the more general theory of directed graphs of large...
On Graphical Regular Representations of Direct Products of Groups.
On graphs all of whose {C₃,T₃}-free arc colorations are kernel-perfect
A digraph D is called a kernel-perfect digraph or KP-digraph when every induced subdigraph of D has a kernel. We call the digraph D an m-coloured digraph if the arcs of D are coloured with m distinct colours. A path P is monochromatic in D if all of its arcs are coloured alike in D. The closure of D, denoted by ζ(D), is the m-coloured digraph defined as follows: V( ζ(D)) = V(D), and A( ζ(D)) = ∪_{i} {(u,v) with colour i: there exists a monochromatic...
On graphs associated to rings.
On graphs containing many subgraphs with the same number of edges
On graphs G for which both g and G̅ are claw-free
Let G be a graph with |V(G)| ≥ 10. We prove that if both G and G̅ are claw-free, then minΔ(G), Δ(G̅) ≤ 2. As a generalization of this result in the case where |V(G)| is sufficiently large, we also prove that if both G and G̅ are -free, then minΔ(G),Δ(G̅) ≤ r(t- 1,t)-1 where r(t-1,t) is the Ramsey number.
On Graphs Whose Energy Does Not Exceed 4
On graphs whose reduced energy does not exceed 3.
On Graphs Whose Second Largest Eigenvalue Does Not Exceed 1
On graphs whose second least eigenvalue is at least
On graphs whose spectral spread does not exceed 4.