arXiv subjects

Vahan Mkrtchyan

Publications and source records attributed to Vahan Mkrtchyan.

28 records · Page 2Linked to original sources

arXiv2019

Normal $5$-edge-colorings of a family of Loupekhine snarks

In a proper edge-coloring of a cubic graph an edge $uv$ is called poor or rich, if the set of colors of the edges incident to $u$ and $v$ contains exactly three or five colors, respectively. An edge-coloring of a graph is normal, if any edge of the graph is either poor or rich. In this note, we show that some snarks constructed by using a method introduced by Loupekhine admit a normal edge-coloring with five colors. The existence of a Berge-Fulkerson Covering for a part of the snarks considered in this paper was recently proved by Manuel and Shanthi (2015). Since the existence of a normal edge-coloring with five colors implies the existence of a Berge-Fulkerson Covering, our main theorem can be viewed as a generalization of their result.

cs.DM↗

arXiv2019

Normal edge-colorings of cubic graphs

A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors having the additional property that when looking at the set of colors assigned to any edge $e$ and the four edges adjacent it, we have either exactly five distinct colors or exactly three distinct colors. We denote by $χ'_{N}(G)$ the smallest $k$, for which $G$ admits a normal $k$-edge-coloring. Normal $k$-edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. More precisely, it is known that proving $χ'_{N}(G)\leq 5$ for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture and then, among others, Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the larger class of all simple cubic graphs (not necessarily bridgeless), some interesting questions naturally arise. For instance, there exist simple cubic graphs, not bridgeless, with $χ'_{N}(G)=7$. On the other hand, the known best general upper bound for $χ'_{N}(G)$ was $9$. Here, we improve it by proving that $χ'_{N}(G)\leq7$ for any simple cubic graph $G$, which is best possible. We obtain this result by proving the existence of specific no-where zero $\mathbb{Z}_2^2$-flows in $4$-edge-connected graphs.

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arXiv2018

On Sylvester Colorings of Cubic Graphs

If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger states that for any bridgeless cubic graph $G$, one has: $P\prec G$. The second author has recently introduced the Sylvester coloring conjecture, which states that for any cubic graph $G$ one has: $S\prec G$. Here $S$ is the Sylvester graph on $10$ vertices. In this paper, we prove the analogue of Sylvester coloring conjecture for cubic pseudo-graphs. Moreover, we show that if $G$ is any connected simple cubic graph $G$ with $G\prec P$, then $G = P$. This implies that the Petersen graph does not admit an $S_{16}$-coloring, where $S_{16}$ is the smallest connected simple cubic graph without a perfect matching. $S_{16}$ has $16$ vertices. %We conjecture that there are infinitely many connected cubic simple graphs which do not admit an %$S_{16}$-coloring. Finally, we obtain $2$ results towards the Sylvester coloring conjecture. The first result states that any cubic graph $G$ has a coloring with edges of Sylvester graph $S$ such that at least $\frac45$ of vertices of $G$ meet the conditions of Sylvester coloring conjecture. The second result states that any claw-free cubic graph graph admits an $S$-coloring. This results is an application of our result on cubic pseudo-graphs.

math.CO↗

arXiv2018

$S_{12}$ and $P_{12}$-colorings of cubic graphs

If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger ($P_{10}$-conjecture) states that for any bridgeless cubic graph $G$, one has: $P_{10}\prec G$. The Sylvester coloring conjecture ($S_{10}$-conjecture) states that for any cubic graph $G$, $S_{10}\prec G$. In this paper, we introduce two new conjectures that are related to these conjectures. The first of them states that any cubic graph with a perfect matching admits an $S_{12}$-coloring. The second one states that any cubic graph $G$ whose edge-set can be covered with four perfect matchings, admits a $P_{12}$-coloring. We call these new conjectures $S_{12}$-conjecture and $P_{12}$-conjecture, respectively. Our first results justify the choice of graphs in $S_{12}$-conjecture and $P_{12}$-conjecture. Next, we characterize the edges of $P_{12}$ that may be fictive in a $P_{12}$-coloring of a cubic graph $G$. Finally, we relate the new conjectures to the already known conjectures by proving that $S_{12}$-conjecture implies $S_{10}$-conjecture, and $P_{12}$-conjecture and $(5,2)$-Cycle cover conjecture together imply $P_{10}$-conjecture. Our main tool for proving the latter statement is a new reformulation of $(5,2)$-Cycle cover conjecture, which states that the edge-set of any claw-free bridgeless cubic graph can be covered with four perfect matchings.

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arXiv2018

On maximum $k$-edge-colorable subgraphs of bipartite graphs

If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the largest among all $k$-edge-colorable subgraphs of $G$. For a graph $G$ and $k\geq 0$, let $ν_{k}(G)$ be the number of edges of a maximum $k$-edge-colorable subgraph of $G$. In 2010 Mkrtchyan et al. proved that if $G$ is a cubic graph, then $ν_2(G)\leq \frac{|V|+2ν_3(G)}{4}$. This result implies that if the cubic graph $G$ contains a perfect matching, in particular when it is bridgeless, then $ν_2(G)\leq \frac{ν_1(G)+ν_3(G)}{2}$. One may wonder whether there are other interesting graph-classes, where a relation between $ν_2(G)$ and $\frac{ν_1(G)+ν_3(G)}{2}$ can be proved. Related with this question, in this paper we show that $ν_{k}(G) \geq \frac{ν_{k-i}(G) + ν_{k+i}(G)}{2}$ for any bipartite graph $G$, $k\geq 0$ and $i=0,1,...,k$.

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arXiv2018

A tight lower bound for the hardness of clutters

A {\it clutter} (or {\it antichain} or {\it Sperner family}) $L$ is a pair $(V,E)$, where $V$ is a finite set and $E$ is a family of subsets of $V$ none of which is a subset of another. Normally, the elements of $V$ are called {\it vertices} of $L$, and the elements of $E$ are called {\it edges} of $L$. A subset $s_e$ of an edge $e$ of a clutter is {\it recognizing} for $e$, if $s_e$ is not a subset of another edge. The {\it hardness} of an edge $e$ of a clutter is the ratio of the size of $e\textrm{'s}$ smallest recognizing subset to the size of $e$. The hardness of a clutter is the maximum hardness of its edges. In this short note we prove a lower bound for the hardness of an arbitrary clutter. Our bound is asymptotically best-possible in a sense that there is an infinite sequence of clutters attaining our bound.

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arXiv2017

Graphs, Disjoint Matchings and Some Inequalities

For $k \geq 1$ and a graph $G$ let $ν_k(G)$ denote the size of a maximum $k$-edge-colorable subgraph of $G$. Mkrtchyan, Petrosyan and Vardanyan proved that $ν_2(G)\geq \frac45\cdot |V(G)|$, $ν_3(G)\geq \frac76\cdot |V(G)|$ for any cubic graph $G$ ~\cite{samvel:2010}. They were also able to show that if $G$ is a cubic graph, then $ν_2(G)+ν_3(G)\geq 2\cdot |V(G)|$ ~\cite{samvel:2014} and $ν_2(G) \leq \frac{|V(G)| + 2\cdot ν_3(G)}{4}$ ~\cite{samvel:2010}. In the first part of the present work, we show that the last two inequalities imply the first two of them. Moreover, we show that $ν_2(G) \geq α\cdot \frac{|V(G)| + 2\cdot ν_3(G)}{4} $, where $α=\frac{16}{17}$, if $G$ is a cubic graph, $α=\frac{20}{21}$, if $G$ is a cubic graph containing a perfect matching, $α=\frac{44}{45}$, if $G$ is a bridgeless cubic graph. We also investigate the parameters $ν_2(G)$ and $ν_3(G)$ in the class of claw-free cubic graphs. We improve the lower bounds for $ν_2(G)$ and $ν_3(G)$ for claw-free bridgeless cubic graphs to $ν_2(G)\geq \frac{35}{36}\cdot |V(G)|$ ($n \geq 48$), $ν_3(G)\geq \frac{43}{45}\cdot |E(G)|$. On the basis of these inequalities we are able to improve the coefficient $α$ for bridgeless claw-free cubic graphs. In the second part of the work, we prove lower bounds for $ν_k(G)$ in terms of $\frac{ν_{k-1}(G)+ν_{k+1}(G)}{2}$ for $k\geq 2$ and graphs $G$ containing at most $1$ cycle. We also present the corresponding conjectures for bipartite and nearly bipartite graphs.

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arXiv2017

Complexity issues in some clustering problems in combinatorial circuits

The modern integrated circuit is one of the most complex products that has been engineered to-date. It continues to grow in complexity as the years progress. As a result, very large-scale integrated (VLSI) circuit design now involves massive design teams employing state-of-the art computer-aided design (CAD) tools. One of the oldest, yet most important CAD problems for VLSI circuits is physical design automation, where one needs to compute the best physical layout of millions to billions of circuit components on a tiny silicon surface \cite{Lim08}. The process of mapping an electronic design to a chip involves a number of physical design stages, one of which is clustering. In this paper, we focus on problems in clustering which are critical for more sustainable chips. The clustering problem in combinatorial circuits alone is a source of multiple models. In particular, we consider the problem of clustering combinatorial circuits for delay minimization, when logic replication is not allowed ({\sc CN}). The problem of delay minimization when logic replication is allowed ({\sc CA}) has been well studied, and is known to be solvable in polynomial-time \cite{Wong1}. However, unbounded logic replication can be quite expensive. Thus, {\sc CN} is an important problem. We show that selected variants of {\sc CN} are {\bf NP-hard}. We also obtain approximability and inapproximability results for these problems. A preliminary version of this paper appeared in \cite{Don15}.

cs.DS↗

arXiv2010

Measures of edge-uncolorability

The resistance $r(G)$ of a graph $G$ is the minimum number of edges that have to be removed from $G$ to obtain a graph which is $Δ(G)$-edge-colorable. The paper relates the resistance to other parameters that measure how far is a graph from being $Δ$-edge-colorable. The first part considers regular graphs and the relation of the resistance to structural properties in terms of 2-factors. The second part studies general (multi-) graphs $G$. Let $r_v(G)$ be the minimum number of vertices that have to be removed from $G$ to obtain a class 1 graph. We show that $\frac{r(G)}{r_v(G)} \leq \lfloor \frac{Δ(G)}{2} \rfloor$, and that this bound is best possible.

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arXiv2010

Bricks and conjectures of Berge, Fulkerson and Seymour

An $r$-graph is an $r$-regular graph where every odd set of vertices is connected by at least $r$ edges to the rest of the graph. Seymour conjectured that any $r$-graph is $r+1$-edge-colorable, and also that any $r$-graph contains $2r$ perfect matchings such that each edge belongs to two of them. We show that the minimum counter-example to either of these conjectures is a brick. Furthermore we disprove a variant of a conjecture of Fan, Raspaud.

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