Robert Šámal

Star Chromatic Index

The star chromatic index χs′(G) of a graph G is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored. We obtain a near-linear upper bound in terms of the maximum degree Δ=Δ(G). Our best lower bound on χs′ in terms of Δ is 2Δ(1+o(1)) valid for complete graphs. We also consider the special case of cubic graphs, for which we show that the star chromatic index lies between 4 and 7 and characterize the graphs attaining the lower bound. The proofs involve a variety of notions from other branches of mathematics and may therefore be of certain independent interest.

Tension continuous maps-Their structure and applications

We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tension-continuous mappings (shortly TT mappings). The existence of a TT mapping induces a (quasi)order on the class of graphs, which seems to be an essential extension of the homomorphism order (studied extensively, see Hell and Nesetril (2004) [10]). In this paper we study the relationship of the homomorphism and TT orders. We stress the similarities and the differences in both deterministic and random settings. Particularly, we prove that TT order is universal and investigate graphs for which homomorphisms and TT mappings coincide (so-called homotens graphs). In the course of our study, we prove a new Ramsey-type theorem, which may be of independent interest. We solve a problem asked in [Matt DeVos, Jaroslav Nesetril, Andre Raspaud, On edge-maps whose inverse preserves flows and tensions, in: J.A. Bondy, J. Fonlupt, J.-L. Fouquet, J.-C. Fournier, J.L. Ramirez Alfonsin (Eds.), Graph Theory in Paris: Proceedings of a Conference in Memory of Claude Berge, in: Trends in Mathematics, Birkhauser, 2006].

Short Cycle Covers of Graphs with Minimum Degree Three

The shortest cycle cover conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with

Fractional covering by cuts

m

Antisymmetric flows and strong oriented coloring of planar graphs

edges can be covered by cycles of total length at most

Cycle-continuous mappings — order structure

7m/5=1.400m

Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings

. We show that every cubic bridgeless graph has a cycle cover of total length at most

A quadratic lower bound for subset sums

34m/21\approx1.619m

Complexity of the cop and robber guarding game

, and every bridgeless graph with minimum degree three has a cycle cover of total length at most

High-girth cubic graphs are homomorphic to the Clebsch graph

44m/27\approx1.630m