Prírodovedecká fakulta

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. In 1978, it was conjectured that ∆(G) + 2 colors suffice for an acyclic edge coloring of every graph G [6]. The conjecture has been verified for several classes of graphs, however, the best known upper bound for as special class as planar graphs are, is ∆ + 12 [2]. In this paper, we study simple planar graphs which need only ∆(G) colors for an acyclic edge coloring. We show that a planar graph with girth g and maximum degree ∆ admits such acyclic edge coloring if g ≥ 12, or g ≥ 8 and ∆ ≥ 4, or g ≥ 7 and ∆ ≥ 5, or g ≥ 6 and ∆ ≥ 6, or g ≥ 5 and ∆ ≥ 10. Our results improve some previously known bounds.

Observability of a graph G is the minimum k for which the edges of G can be properly coloured with k colours in such a way that colour sets of vertices of G (sets of colours of their incident edges) are pairwise distinct. It is shown that limn--oo obs~Qn~ = 1 + q* where n q* = 0.293815... is the unique solution of the equation (x + 1)~+1 = 2x • in the interval (0, c~).

The vertex-distinguishing index χ s (G) of a graph G is the minimum number of colours required to properly colour the edges of G in such a way that any two vertices are incident with different sets of colours. We consider this parameter for some regular graphs. Moreover, we prove that for any graph, the value of this invariant is not changed if the colouring above is, in addition, required to be equitable.

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree ∆ has a strong edge coloring with at most 4 ∆ + 4 colors. We show that 3 ∆ + 6 colors suffice if the graph has girth 6, and 3 ∆ colors suffice if the girth is at least 7. Moreover, we show that cubic planar graphs with girth at least 6 can be strongly edge colored with at most 9 colors.

For a given graph H and n ? 1; let f(n;H) denote the maximum number m for which it is possible to colour the edges of the complete graph Kn with m colours in such a way that each subgraph H in Kn has at least two edges of the same colour. Equivalently, any edge-colouring of Kn with at least rb(n;H) = f(n;H)+1 colours contains a rainbow copy of H: The numbers f(n;H) and rb(Kn;H) are called anti-ramsey numbers and rainbow numbers, respectively. In this paper we will classify the rainbow number for a given graph H with respect to its cyclomatic number. Let H be a graph of order p >= 4 and cyclomatic number v(H) >= 2: Then rb(Kn;H) cannot be bounded from above by a function which is linear in n: If H has cyclomatic number v(H) = 1; then rb(Kn;H) is linear in n: We will compute all rainbow numbers for the bull B; which is the unique graph with 5 vertices and degree sequence (1; 1; 2; 3; 3): Furthermore, we will compute some rainbow numbers for the diamond D = K_4 - e; for K_2;3; an...

A strong edge coloring of a graph G is a proper edge coloring in which each color class is an induced matching of G. In 1993, Brualdi and Quinn Massey [3] proposed a conjecture that every bipartite graph without 4-cycles and with the maximum degrees of the two partite sets 2 and ∆ admits a strong edge coloring with at most ∆ + 2 colors. We prove that this conjecture holds for such graphs with ∆ = 3. We also confirm the conjecture proposed by Faudree et al. for subcubic bipartite graphs. NOTE: After publishing this paper on ArXiv, we have been notified that an equivalent result has already been proven by Maydanskiy [9] in the scope of incidence colorings. Moreover, the latter result has been verified in a little stronger form in by Wu and Lin.

We study 3-valent maps M n (p, q) consisting of a ring of n q-gons whose the inner and outer domains are filled by p-gons, for p, q ≥ 3. We describe a domain in the space of parameters p, q, and n, for which such a map may exist. With four infinite sequences of maps -prisms M p (p ≥ 3, 4), M 4 (4, q ≥ 4), M 4 (5, 5t+2 ≥ 7), M 4 (5, 5t+3 ≥ 8), we give 26 sporadic ones. The maps whose p-gons form two paths are first two infinite sequences and 5 maps: M 28 (7, 5), M 12 (6, 5), M 10 (5, 6), M 20 (5, 7), M 2 (3, 6).

A star edge coloring of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this paper we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.

Integers have many interesting properties. In this paper it will be shown that, for an arbitrary nonconstant arithmetic progression {a n }™ =l of positive integers (denoted by N), either {a n }™ =l contains infinitely many palindromic numbers or else 10|a w for every n GN. (This result is a generalization of the theorem concerning the existence of palindromic multiples, cf. [2].) More generally, for any number system base b, a nonconstant arithmetic progression of positive integers contains infinitely many palindromic numbers if and only if there exists a member of the progres-sion not divisible by b. WHAT IS A PALINDROMIC NUMBER? A positive integer is said to be a (decadic) palindromic number or, shortly, a palindrome if its leftmost digit is the same as its rightmost digit, its second digit from the left is equal to its second digit from the right, and so on. For example, 33, 142505241, and 6 are palindromic numbers. More precisely, let d k d k _ l ...d l dQ be a usual decadic exp...

The point-distinguishing chromatic index of a graph represents the minimum number of colours in its edge colouring such that each vertex is distinguished by the set of colours of edges incident with it. Asymptotic information on jumps of the point-distinguishing chromatic index of K n,n is found.

A subgraph of a plane graph is light if each of its vertices has a small degree in the entire graph. Consider the class ~¢-(5) of plane triangulations of minimum degree 5. It is known that each G C,Y-(5) contains a light triangle. From a recent result of Jendrol' and Madaras the existence of light cycles C4 and C5 in each G C.~(5) follows. We prove here that each G E.Y-(5) contains also light cycles Q,,CT, C~ and C~ such that every vertex is of degree at most 11, 17,29 and 41, respectively. Moreover, we prove that no cycle C~ with k ~> 11 is light in the class #-(5).

Let P and Q be additive and hereditary graph properties and let r, s be integers such that r ≥ s. Then an r s-fractional (P, Q)-total coloring of a finite graph G = (V, E) is a mapping f , which assigns an s-element subset of the set {1, 2,. .. , r} to each vertex and each edge, moreover, for any color i all vertices of color i induce a subgraph with property P, all edges of color i induce a subgraph with property Q and vertices and incident edges have been assigned disjoint sets of colors. The minimum ratio of an r s-fractional (P, Q)-total coloring of G is called fractional (P, Q)-total chromatic number χ ′′ f,P,Q (G) = r s. We show in this paper that χ ′′ f,P,Q of a graph G with o(V (G)) vertex orbits and o(E(G)) edge orbits can be found as a solution of a linear program with integer coefficients which consists only of o(V (G)) + o(E(G)) inequalities.