arXiv subjects

Gagik N. Vardanyan

Publications and source records attributed to Gagik N. Vardanyan.

4 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

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↗

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↗

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↗