Zdzisław Skupień

Smallest Sets of Longest Paths with Empty Intersection

It is shown that, for every integer v v – 1 longest paths have a vertex in common. Moreover, connected graphs having seven or five minimal sets of longest paths (longest cycles) with empty intersection are presented. A 26-vertex 2-connected graph whose longest paths have empty intersection is exhibited.

Local structures in plane maps and distance colourings

Abstract A structural result on normal plane maps is presented. It strengthens a result by Borodin which is related to Kotzigs and Lebesgues classical results. Then the t-distance chromatic number of a planar graph G with maximum degree Δ ( G )⩽ D where D ⩾8 is proved to be bounded above by a polynomial in D of degree t −1.

Decompositions of multigraphs into parts with two edges

Given a family F of multigraphs without isolated vertices, a multigraph M is called F-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of F . We present necessary and sufficient conditions for the existence of such decompositions if F comprises two multigraphs from the set consisting of a 2-cycle, a 2-matching and a path with two edges.

Degrees in Homogeneously Traceable Graphs

It is proved that if G is a non-Hamiltonian homogeneously traceable graph of order n ⩾ 3, then for each vertex ν of G there exists a ν - w Hamiltonian path whose end-vertices v and w have the sum of degrees less than or equal to n-2. Hence Δ(G) + δ(G)⩽n -2. Open related problems are stated.

Hamiltonicity of planar cubic multigraphs

Hamiltonicity of connected cubic planar general graphs G is characterized in terms of partitioning any dual graph G* into two trees. Thus tree-tree triangulations become involved. The related Stein theorem is corrected and extended. Moreover, it follows that a maximal planar graph G can be partitioned into two forests if and only if G can be partitioned into two trees.

The complete grapht-packings andt-coverings

The aim of this paper is to introduce the notions of floor and ceiling functions related to graph edge-decompositions intot mutually isomorphic parts. It is shown that, given any natural numbert, uniform and extremal t-packing andt-covering exist for each complete graph and each complete bipartite graph. Extremal in this context means that both a remainder and a surplus are absolutely minimum. In proofs, decompositions of multigraphs into matchings are involved. Open problems and conjectures are stated.

BCH codes and distance multi- or fractional colorings in hypercubes asymptotically

The main result is a short and elementary proof for the authors exact asymptotic results on distance chromatic parameters (both number and index) in hypercubes. Moreover, the results are extended to those on fractional distance chromatic parameters and on distance multi-colorings. Inspiration comes from radio frequencies allocation problem. The basic idea is the observation that binary primitive narrow-sense BCH codes or their shortenings have size asymptotically within a constant factor below the largest possible size, A(n,d), among all binary codes of the same length, n, and the same minimum distance, d, as n->~ while d is constant. Also a lower bound in terms of A(n,d) is obtained for B(n,d), the largest size among linear binary codes.

On traceability and 2-factors in claw-free graphs

If G is a claw-free graph of sufficiently large order n, satisfying a degree condition σk > n+k2−4k+7 (where k is an arbitrary constant), then G has a 2-factor with at most k − 1 components. As a second main result, we present classes of graphs C1, . . . , C8 such that every sufficiently large connected claw-free graph satisfying degree condition σ6(k) > n + 19 (or, as a corollary, δ(G) > n+19 6 ) either belongs to ∪i=1Ci or is traceable.

Exponential Constructions of some Nonhamiltonian Minima

Exponentially many n-vertex minimum nonhamiltonian (A) homogeneously traceable graphs and (B) bihomogeneously traceable oriented graphs are constructed. An analog of Sylvesters result on numerical semigroups is used.

Majorization and the minimum number of dominating sets

A recent Wagner result, which characterizes graphs without isolates and with a minimum number of dominating sets, is based on an earlier characterization restricted to trees. A self-contained proof of both characterizations has been found. Each graph in question is proved to include a union of disjoint stars as a spanning subgraph which is induced by pendant edges of the graph. Majorization among numerical multisets which represent distribution of vertices among stars helps to see that the solution is equipartite. Continuization yields the optimal size of stars. Additional characterizations limited to graphs with upper-/lower-bounded domination number are obtained. A linear-time algorithm for partitioning a tree into minimal number of stars is presented. Concluding remarks indicate how majorization applied to disjoint unions of subdivided stars leads to a characterization of graphs with nontrivial minimum number of totally dominating sets. Amazingly, these graphs of order different from 38 make up a well-defined class of graphs with a maximum number of efficient dominating sets.