Ján Mazák

Optimal acyclic edge-coloring of cubic graphs

An acyclic edge-coloring of a graph is a proper edge-coloring such that the subgraph induced by the edges of any two colors is acyclic. The acyclic chromatic index of a graph G is the smallest number of colors in an acyclic edge-coloring of G. We prove that the acyclic chromatic index of a connected cubic graph G is 4, unless G is K4 or K3,3; the acyclic chromatic index of K4 and K3,3 is 5. This result has previously been published by Fiamcik, but his published proof was erroneous.

Cubic Graphs with Given Circular Chromatic Index

For every rational number

Weak oddness as an approximation of oddness and resistance in cubic graphs

r

Simple cubic graphs with no short traveling salesman tour

such that

On even cycle decompositions of 4-regular line graphs

3<r<3+1/3

An algorithm for optimal acyclic edge-colouring of cubic graphs

, we construct an infinite family of cubic graphs with circular chromatic index

Cubic Graphs with Large Circumference Deficit

r

Non-trivial snarks with given circular chromatic index

. Our construction disproves a conjecture of Zhu that there is no bounded increasing infinite sequence in the set of circular chromatic indices of graphs [Topics in Discrete Mathematics, Dedicated to Jarik Nesbetrbil on the Occasion of his 60th Birthday, Algorithms Combin. 26, M. Klazar, J. Kratochvil, M. Loebl, J. Matousek, R. Thomas, and P. Valtr, eds., Springer, Berlin, 2006, pp. 497-550]. We also describe a sequence of graphs whose circular chromatic indices converge to

Snarks with large oddness and small number of vertices

3+1/3

Degenerated induced subgraphs of planar graphs

from above.