arXiv subjects

Vahan V. Mkrtchyan

Publications and source records attributed to Vahan V. Mkrtchyan.

12 recordsLinked to original sources

arXiv2018

On two consequences of Berge-Fulkerson conjecture

The classical Berge-Fulkerson conjecture states that any bridgeless cubic graph $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to two of the perfect matchings from the list. In this short note, we discuss two statements that are consequences of this conjecture. We show that the first statement is equivalent to Fan-Raspaud conjecture. We also show that the smallest counter-example to the second one is a cyclically $4$-edge-connected cubic graph.

cs.DM↗

arXiv2015

The hardness of the independence and matching clutter of a graph

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. Usually, 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 called {\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. We study the hardness of clutters arising from independent sets and matchings of graphs.

cs.DM↗

arXiv2015

On strongly spanning $k$-edge-colorable subgraphs

A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a graph-parameter $sp(G)$, that coincides with the smallest $k$ that a graph $G$ has a strongly spanning maximum $k$-edge-colorable subgraph. Our first result offers some alternative definitions of $sp(G)$. Next, we show that $Δ(G)$ is an upper bound for $sp(G)$, and then we characterize the class of graphs $G$ that satisfy $sp(G)=Δ(G)$. Finally, we prove some bounds for $sp(G)$ that involve well-known graph-theoretic parameters.

cs.DM↗

arXiv2015

Graph Theory

This book is based on Graph Theory courses taught by P.A. Petrosyan, V.V. Mkrtchyan and R.R. Kamalian at Yerevan State University.

math.CO↗

arXiv2012

Equitable Colorings of Corona Multiproducts of Graphs

A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the equitable chromatic number of $G$ and denoted $χ_{=}(G)$. It is known that this problem is NP-hard in general case and remains so for corona graphs. In "Equitable colorings of Cartesian products of graphs" (2012) Lin and Chang studied equitable coloring of Cartesian products of graphs. In this paper we consider the same model of coloring in the case of corona products of graphs. In particular, we obtain some results regarding the equitable chromatic number for $l$-corona product $G \circ ^l H$, where $G$ is an equitably 3- or 4-colorable graph and $H$ is an $r$-partite graph, a path, a cycle or a complete graph. Our proofs are constructive in that they lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of $G$. Moreover, we confirm Equitable Coloring Conjecture for corona products of such graphs. This paper extends our results from \cite{hf}.

cs.DM↗

arXiv2012

On disjoint matchings in cubic graphs: maximum 2- and 3-edge-colorable subgraphs

We show that any $2-$factor of a cubic graph can be extended to a maximum $3-$edge-colorable subgraph. We also show that the sum of sizes of maximum $2-$ and $3-$edge-colorable subgraphs of a cubic graph is at least twice of its number of vertices. Finally, for a cubic graph $G$, consider the pairs of edge-disjoint matchings whose union consists of as many edges as possible. Let $H$ be the largest matching among such pairs. Let $M$ be a maximum matching of $G$. We show that 9/8 is a tight upper bound for $|M|/|H|$.

cs.DM↗

arXiv2012

A remark on Petersen coloring conjecture of Jaeger

If $G$ and $H$ are two cubic graphs, then we write $H\prec G$, if $G$ admits 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 $f(\partial_G(x))=\partial_H(y)$. Let $P$ and $S$ be the Petersen graph and the Sylvester graph, respectively. In this paper, we introduce the Sylvester coloring conjecture. Moreover, we show that if $G$ is a connected bridgeless cubic graph with $G\prec P$, then $G=P$. Finally, if $G$ is a connected cubic graph with $G\prec S$, then $G=S$.

cs.DM↗

arXiv2011

On upper bounds for parameters related to construction of special maximum matchings

For a graph $G$ let $L(G)$ and $l(G)$ denote the size of the largest and smallest maximum matching of a graph obtained from $G$ by removing a maximum matching of $G$. We show that $L(G)\leq 2l(G),$ and $L(G)\leq (3/2)l(G)$ provided that $G$ contains a perfect matching. We also characterize the class of graphs for which $L(G)=2l(G)$. Our characterization implies the existence of a polynomial algorithm for testing the property $L(G)=2l(G)$. Finally we show that it is $NP$-complete to test whether a graph $G$ containing a perfect matching satisfies $L(G)=(3/2)l(G)$.

cs.DM↗

arXiv2011

Maximum $Δ$-edge-colorable subgraphs of class II graphs

A graph $G$ is class II, if its chromatic index is at least $Δ+1$. Let $H$ be a maximum $Δ$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of maximum $Δ$-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with $Δ\geq3$ can be extended to a maximum $Δ$-edge-colorable subgraph. Simple graphs have a maximum $Δ$-edge-colorable subgraph such that the complement is a matching. Furthermore, a maximum $Δ$-edge-colorable subgraph of a simple graph is always class I.

cs.DM↗

arXiv2011

A note on triangle-free graphs

We show that if $G$ is a simple triangle-free graph with $n\geq 3$ vertices, without a perfect matching, and having a minimum degree at least $\frac{n-1}{2}$, then $G$ is isomorphic either to $C_5$ or to $K_{\frac{n-1}{2},\frac{n+1}{2}}$.

cs.DM↗

arXiv2010

On disjoint matchings in cubic graphs

For $i=2,3$ and a cubic graph $G$ let $ν_{i}(G)$ denote the maximum number of edges that can be covered by $i$ matchings. We show that $ν_{2}(G)\geq {4/5}| V(G)| $ and $ν_{3}(G)\geq {7/6}| V(G)| $. Moreover, it turns out that $ν_{2}(G)\leq \frac{|V(G)|+2ν_{3}(G)}{4}$.

cs.DM↗

arXiv2008

5-cycles and the Petersen graph

We show that if G is a connected bridgeless cubic graph whose every 2-factor is comprised of cycles of length five then G is the Petersen graph.

cs.DM↗