Corpus ID: 15177037

# Graphs, Disjoint Matchings and Some Inequalities

@article{Hambardzumyan2015GraphsDM,
title={Graphs, Disjoint Matchings and Some Inequalities},
author={Lianna Hambardzumyan and V. Mkrtchyan},
journal={arXiv: Discrete Mathematics},
year={2015}
}
• Published 2015
• Computer Science, Mathematics
• arXiv: Discrete Mathematics
For $k \geq 1$ and a graph $G$ let $\nu_k(G)$ denote the size of a maximum $k$-edge-colorable subgraph of $G$. Mkrtchyan, Petrosyan and Vardanyan proved that $\nu_2(G)\geq \frac45\cdot |V(G)|$, $\nu_3(G)\geq \frac76\cdot |V(G)|$ for any cubic graph $G$ ~\cite{samvel:2010}. They were also able to show that if $G$ is a cubic graph, then $\nu_2(G)+\nu_3(G)\geq 2\cdot |V(G)|$ ~\cite{samvel:2014} and $\nu_2(G) \leq \frac{|V(G)| + 2\cdot \nu_3(G)}{4}$ ~\cite{samvel:2010}. In the first part of the… Expand
1 Citations

#### Figures and Topics from this paper

On maximum k-edge-colorable subgraphs of bipartite graphs
• Mathematics, Computer Science
• Discret. Appl. Math.
• 2019
The result implies that if the cubic graph G contains a perfect matching, in particular when it is bridgeless, then ν 2 ( G) ≤ ν 1 ( G ) + ν 3 ( G ), which is the number of edges of a maximum k -edge-colorable subgraph of G. Expand

#### References

SHOWING 1-10 OF 24 REFERENCES
Maximum Matchings in Regular Graphs of High Girth
• Mathematics, Computer Science
• Electron. J. Comb.
• 2007
This note shows that G has a maximum matching which includes all but an exponentially small fraction of the vertices, O((d-1)^{-g/2})$, where the number of unmatched vertices is at most$n/n_0(d,g)$. Expand Beyond the Vizing's Bound for at Most Seven Colors • Mathematics, Computer Science • SIAM J. Discret. Math. • 2014 Lower bounds on the size of subgraphs of G that can be colored with at most$\Delta +1$colors are studied and a new bound for subcubic multigraphs not isomorphic to$K_3$with one edge doubled is shown. Expand Tight Lower Bounds on the Size of a Maximum Matching in a Regular Graph • Mathematics, Computer Science • Graphs Comb. • 2007 Tight lower bounds on the size of a maximum matching in a regular graph of order n and α′(G) are studied, which show that if k is even, then α' (G) \ge \min \left(k^3-k^2-2) \, n - 2k + 2}{2(k-3-3k)} . Expand On Parsimonious Edge-Colouring of Graphs with Maximum Degree Three • Mathematics, Computer Science • Graphs Comb. • 2013 These results are obtained by using structural properties of the so called δ-minimum edge colourings for graphs with maximum degree 3, with the exception of a graph on 5 vertices. Expand Approximating the maximum 3-edge-colorable subgraph problem • R. Rizzi • Computer Science, Mathematics • Discret. Math. • 2009 An improved approximation algorithm is obtained for the Maximum k-Edge-Colorable Subgraph Problem, achieving approximation ratio of 6/7, and better bounds are obtained for graphs of higher odd girth. Expand Measurements of edge-uncolorability • E. Steffen • Computer Science, Mathematics • Discret. Math. • 2004 Abstract Cubic bridgeless graphs with chromatic index four are called uncolorable. We introduce parameters measuring the uncolorability of those graphs and relate them to each other. For k =2,3, letExpand On disjoint matchings in cubic graphs • Computer Science, Mathematics • Discret. Math. • 2010 It turns out that @n"2 (G)@?|V(G)|+2@n"3(G)4.4 is the maximum number of edges that can be covered by i matchings for a cubic graph G. Expand On disjoint matchings in cubic graphs: Maximum 2-edge-colorable and maximum 3-edge-colorable subgraphs • Computer Science, Mathematics • Discret. Appl. Math. • 2014 It is shown that any 2-factor of a cubic graph can be extended to a maximum 3-edge-colorable subgraph and that 9 / 8 is a tight upper bound for that subgraph. Expand Perfect Matchings in Claw-free Cubic Graphs • Sang-il Oum • Mathematics, Computer Science • Electron. J. Comb. • 2011 It is proved that every claw-free cubic$n-vertex graph with no cutedge has more than $2^{n/12}$ perfect matchings, thus verifying the conjecture for claw- free graphs. Expand
Introduction to Graph Theory
1. Fundamental Concepts. What Is a Graph? Paths, Cycles, and Trails. Vertex Degrees and Counting. Directed Graphs. 2. Trees and Distance. Basic Properties. Spanning Trees and Enumeration.Expand