Ararat Harutyunyan

Two results on the digraph chromatic number

It is known (Bollobas (1978) [4]; Kostochka and Mazurova (1977) [12]) that there exist graphs of maximum degree @D and of arbitrarily large girth whose chromatic number is at least c@D/log@D. We show an analogous result for digraphs where the chromatic number of a digraph D is defined as the minimum integer k so that V(D) can be partitioned into k acyclic sets, and the girth is the length of the shortest cycle in the corresponding undirected graph. It is also shown, in the same vein as an old result of Erdos (1962) [5], that there are digraphs with arbitrarily large chromatic number where every large subset of vertices is 2-colorable.

Gallai's Theorem for List Coloring of Digraphs

A classical theorem of Gallai states that in every graph that is critical for k-colorings, the vertices of degree

Edge-Partitioning a Graph into Paths: Beyond the Barát-Thomassen Conjecture

k-1

A proof of the Barát-Thomassen conjecture

induce a tree-like graph whose blocks are either complete graphs or cycles of odd length. We provide a generalization to colorings and list colorings of digraphs, where some new phenomena arise. In particular, the problem of list coloring digraphs with the lists at each vertex v having

Strong edge-colouring of sparse planar graphs

\min\{d^{+}(v),d^{-}(v)\}

Planar Digraphs of Digirth Five Are 2-Colorable

colors turns out to be NP-hard.

The complexity of tropical graph homomorphisms

In 2006, Barát and Thomassen conjectured that there is a function f such that, for every fixed tree T with t edges, every f(t)-edge-connected graph with its number of edges divisible by t has a partition of its edges into copies of T. This conjecture was recently verified by the current authors and Merker [1].We here further focus on the path case of the Barát-Thomassen conjecture. Before the aforementioned general proof was announced, several successive steps towards the path case of the conjecture were made, notably by Thomassen [11,12,13], until this particular case was totally solved by Botler, Mota, Oshiro andWakabayashi [2]. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function f such that every 24-edge-connected graph with minimum degree f(t) has an edge-partition into paths of length t whenever t divides the number of edges. We also show that 24 can be dropped to 4 when the graph is eulerian.

Some bounds on global alliances in trees

The Barat-Thomassen conjecture asserts that for every tree T on m edges, there exists a constant k T such that every k T-edge-connected graph with size divisible by m can be edge-decomposed into copies of T. So far this conjecture has only been verified when T is a path or when T has diameter at most 4. Here we prove the full statement of the conjecture.

UNIQUELY D-COLOURABLE DIGRAPHS WITH LARGE GIRTH

A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree Δ has a strong edge-colouring with at most 4 Δ + 4 colours. We show that 3 Δ + 1 colours suffice if the graph has girth 6, and 4 Δ colours suffice if Δ ? 7 or the girth is at least 5. In the last part of the paper, we raise some questions related to a long-standing conjecture of Vizing on proper edge-colouring of planar graphs.

A fast algorithm for powerful alliances in trees

Neumann-Lara (1985) and Skrekovski conjectured that every planar digraph with digirth at least three is 2-colorable, meaning that the vertices can be 2-colored without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least five is 2-colorable. The result also holds in the setting of list colorings.