arXiv subjects

Ararat Harutyunyan

Publications and source records attributed to Ararat Harutyunyan.

30 records · Page 2Linked to original sources

arXiv2018

The complexity of tropical graph homomorphisms

A tropical graph $(H,c)$ consists of a graph $H$ and a (not necessarily proper) vertex-colouring $c$ of $H$. Given two tropical graphs $(G,c_1)$ and $(H,c)$, a homomorphism of $(G,c_1)$ to $(H,c)$ is a standard graph homomorphism of $G$ to $H$ that also preserves the vertex-colours. We initiate the study of the computational complexity of tropical graph homomorphism problems. We consider two settings. First, when the tropical graph $(H,c)$ is fixed; this is a problem called $(H,c)$-COLOURING. Second, when the colouring of $H$ is part of the input; the associated decision problem is called $H$-TROPICAL-COLOURING. Each $(H,c)$-COLOURING problem is a constraint satisfaction problem (CSP), and we show that a complexity dichotomy for the class of $(H,c)$-COLOURING problems holds if and only if the Feder-Vardi Dichotomy Conjecture for CSPs is true. This implies that $(H,c)$-COLOURING problems form a rich class of decision problems. On the other hand, we were successful in classifying the complexity of at least certain classes of $H$-TROPICAL-COLOURING problems.

cs.DS↗

arXiv2017

Coloring tournaments: from local to global

The \emph{chromatic number} of a directed graph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class of $D$ induces an acyclic subdigraph. Thus, the chromatic number of a tournament $T$ is the minimum number of transitive subtournaments which cover the vertex set of $T$. We show in this paper that tournaments are significantly simpler than graphs with respect to coloring. Indeed, while undirected graphs can be altogether "locally simple" (every neighborhood is a stable set) and have large chromatic number, we show that locally simple tournaments are indeed simple. In particular, there is a function $f$ such that if the out-neighborhood of every vertex in a tournament $T$ has chromatic number at most $c$, then $T$ has chromatic number at most $f(c)$. This answers a question of Berger et al.

math.CO↗

arXiv2016

A Proof of the Barát-Thomassen Conjecture

The Barát-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.

math.CO↗

arXiv2016

Edge-partitioning a graph into paths: beyond the Barát-Thomassen conjecture

The Barát-Thomassen conjecture asserts that there is a function $f$ such that for every fixed tree $T$ with $t$ edges, every graph which is $f(t)$-edge-connected with its number of edges divisible by $t$ has a partition of its edges into copies of $T$. This has been proved in the case of paths of length $2^k$ by Thomassen, and recently shown to be true for all paths by Botler, Mota, Oshiro and Wakabayashi. 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.

math.CO↗

arXiv2016

Disjoint cycles of different lengths in graphs and digraphs

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider stronger situations, and exhibit degree bounds (some of which are best possible) when e.g. the graph is triangle-free, or the $k$ cycles are requested to have different lengths congruent to some values modulo some $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for regular digraphs and digraphs of small order.

math.CO↗

arXiv2015

List coloring digraphs

The dichromatic number $\vecχ(D)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this paper, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's Conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as $Ω(\log_2 n)$. We finally give a Brooks-type upper bound on the list dichromatic number of digon-free digraphs.

math.CO↗

arXiv2014

Strong edge-colouring of sparse planar graphs

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 $4Δ+4$ colours. We show that $3Δ+1$ colours suffice if the graph has girth 6, and $4Δ$ colours suffice if $Δ\geq 7$ 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.

cs.DM↗

arXiv2014

Planar digraphs of digirth five are 2-colorable

Neumann-Lara (1985) and Škrekovski conjectured that every planar digraph with digirth at least three is 2-colorable. 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.

math.CO↗

arXiv2011

Strengthened Brooks Theorem for digraphs of girth three

Brooks' Theorem states that a connected graph $G$ of maximum degree $Δ$ has chromatic number at most $Δ$, unless $G$ is an odd cycle or a complete graph. A result of Johansson (1996) shows that if $G$ is triangle-free, then the chromatic number drops to $O(Δ/ \log Δ)$. In this paper, we derive a weak analog for the chromatic number of digraphs. We show that every (loopless) digraph $D$ without directed cycles of length two has chromatic number $χ(D) \leq (1-e^{-13}) \tildeΔ$, where $\tildeΔ$ is the maximum geometric mean of the out-degree and in-degree of a vertex in $D$, when $\tildeΔ$ is sufficiently large. As a corollary it is proved that there exists an absolute constant $α< 1$ such that $χ(D) \leq α(\tildeΔ + 1)$ for every $\tildeΔ > 2$.

math.CO↗

arXiv2011

Two results on the digraph chromatic number

It is known (Bollobás (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $Δ$ and of arbitrarily large girth whose chromatic number is at least $c Δ/ \log Δ$. 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), that there are digraphs with arbitrarily large chromatic number where every large subset of vertices is 2-colorable.

math.CO↗

arXiv2011

Planar graphs have exponentially many 3-arboricities

It is well-known that every planar or projective planar graph can be 3-colored so that each color class induces a forest. This bound is sharp. In this paper, we show that there are in fact exponentially many 3-colorings of this kind for any (projective) planar graph. The same result holds in the setting of 3-list-colorings.

math.CO↗

arXiv2011

Uniquely D-colourable digraphs with large girth

Let C and D be digraphs. A mapping $f:V(D)\to V(C)$ is a C-colouring if for every arc $uv$ of D, either $f(u)f(v)$ is an arc of C or $f(u)=f(v)$, and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number $r\geq 1$, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.

math.CO↗