scholarly journals Short Cycle Covers of Graphs with Minimum Degree Three

2010 ◽  
Vol 24 (1) ◽  
pp. 330-355 ◽  
Author(s):  
Tomáš Kaiser ◽  
Daniel Král' ◽  
Bernard Lidický ◽  
Pavel Nejedlý ◽  
Robert Šámal
Keyword(s):  
Minimum Degree ◽  
Short Cycle ◽  
Cycle Covers

scholarly journals Short Cycle Covers of Graphs with at Most 77% Vertices of Degree Two

10.37236/9284 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Anna Kompišová ◽  
Robert Lukot'ka
Keyword(s):  
Double Cover ◽  
Short Cycle ◽  
Cycle Cover ◽  
M 10 ◽  
Cycle Covers

Let $G$ be a bridgeless multigraph with $m$ edges and $n_2$ vertices of degree two and let $cc(G)$ be the length of its shortest cycle cover. It is known that if $cc(G) < 1.4m$ in bridgeless graphs with $n_2 \le m/10$, then the Cycle Double Cover Conjecture holds. Fan (2017)  proved that if $n_2 = 0$, then $cc(G) < 1.6258m$ and $cc(G) < 1.6148m$ provided that $G$ is loopless; morever, if $n_2 \le m/30$, then $cc(G) < 1.6467m$. We show that for a bridgeless multigraph with $m$ edges and $n_2$ vertices of degree two, $cc(G) < 1.6148m + 0.0741n_2$. Therefore, if $n_2=0$, then $cc(G) < 1.6148m$ even if $G$ has loops; if $n_2 \le m/30$, then $cc(G) < 1.6173m$; and if $n_2 \le m/10$, then $cc(G) < 1.6223|E(G)|$. Our improvement is obtained by randomizing Fan's construction.

Cubic Graphs with No Short Cycle Covers

2021 ◽  
Vol 35 (3) ◽  
pp. 2223-2233
Author(s):  
Edita Máčajová ◽  
Martin Škoviera
Keyword(s):  
Cubic Graphs ◽  
Short Cycle ◽  
Cycle Covers

scholarly journals Stability for Vertex Cycle Covers

10.37236/5185 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
József Balogh ◽  
Frank Mousset ◽  
Jozef Skokan
Keyword(s):  
Minimum Degree ◽  
Independent Set ◽  
Stability Result ◽  
Natural Generalization ◽  
Regularity Lemma ◽  
Property A ◽  
Vertex Graph ◽  
Szemeredi's Regularity Lemma ◽  
Cycle Covers ◽  
Positive Integers

In 1996 Kouider and Lonc proved the following natural generalization of Dirac's Theorem: for any integer $k\geq 2$, if $G$ is an $n$-vertex graph with minimum degree at least $n/k$, then there are $k-1$ cycles in $G$ that together cover all the vertices.This is tight in the sense that there are $n$-vertex graphs that have minimum degree $n/k-1$ and that do not contain $k-1$ cycles with this property. A concrete example is given by $I_{n,k} = K_n\setminus K_{(k-1)n/k+1}$ (an edge-maximal graph on $n$ vertices with an independent set of size $(k-1)n/k+1$). This graph has minimum degree $n/k-1$ and cannot be covered with fewer than $k$ cycles. More generally, given positive integers $k_1,\dotsc,k_r$ summing to $k$, the disjoint union $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$ is an $n$-vertex graph with the same properties.In this paper, we show that there are no extremal examples that differ substantially from the ones given by this construction. More precisely, we obtain the following stability result: if a graph $G$ has $n$ vertices and minimum degree nearly $n/k$, then it either contains $k-1$ cycles covering all vertices, or else it must be close (in ‘edit distance') to a subgraph of $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$, for some sequence $k_1,\dotsc,k_r$ of positive integers that sum to $k$.Our proof uses Szemerédi's Regularity Lemma and the related machinery.

Short cycle covers of graphs and nowhere-zero flows

2011 ◽  
Vol 68 (4) ◽  
pp. 340-348 ◽  
Author(s):  
Edita Máčajová ◽  
André Raspaud ◽  
Michael Tarsi ◽  
Xuding Zhu
Keyword(s):  
Short Cycle ◽  
Cycle Covers

scholarly journals Short cycle covers and the cycle double cover conjecture

1992 ◽  
Vol 56 (2) ◽  
pp. 197-204 ◽  
Author(s):  
Ury Jamshy ◽  
Michael Tarsi
Keyword(s):  
Double Cover ◽  
Short Cycle ◽  
Cycle Covers

Short cycle covers of cubic graphs

1994 ◽  
Vol 18 (2) ◽  
pp. 131-141 ◽  
Author(s):  
Genghua Fan
Keyword(s):  
Cubic Graphs ◽  
Short Cycle ◽  
Cycle Covers

Short Cycle Covers on Cubic Graphs by Choosing a 2-Factor

2016 ◽  
Vol 30 (4) ◽  
pp. 2086-2106 ◽  
Author(s):  
Barbora Candráková ◽  
Robert Lukoťka
Keyword(s):  
Cubic Graphs ◽  
Short Cycle ◽  
Cycle Covers

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

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