scholarly journals Long Cycles in t-Tough Graphs with t > 1

2019 ◽  
pp. 39-56
Author(s):  
Zhora Nikoghosyan
Keyword(s):  
Minimum Degree ◽  
Petersen Graph ◽  
Long Cycles

It is proved that if G is a t-tough graph of order n and minimum degree δ with t > 1, then either G has a cycle of length at least min{n, 2δ + 4} or G is the Petersen graphIt is proved that if G is a t-tough graph of order n and minimum degree δ with t > 1, then either G has a cycle of length at least min{n, 2δ + 4} or G is the Petersen graph

scholarly journals Long cycles in subgraphs with prescribed minimum degree

1991 ◽  
Vol 97 (1-3) ◽  
pp. 69-81 ◽  
Author(s):  
L. Caccetta ◽  
K. Vijayan
Keyword(s):  
Minimum Degree ◽  
Long Cycles

scholarly journals Long cycles in hypercubes with distant faulty vertices

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Petr Gregor ◽  
Riste Škrekovski
Keyword(s):  
Linear Code ◽  
Minimum Distance ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
International Audience ◽  
Long Cycles

Graphs and Algorithms International audience In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.

Excluding Minors in Nonplanar Graphs of Girth at Least Five

2000 ◽  
Vol 9 (6) ◽  
pp. 573-585 ◽  
Author(s):  
ROBIN THOMAS ◽  
JAN McDONALD THOMSON
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
The Other ◽  
Petersen Graph ◽  
Regular Graphs ◽  
Nonplanar Graph ◽  
A Minor

A graph G is quasi 4-connected if it is simple, 3-connected, has at least five vertices, and for every partition (A, B, C) of V(G) either [mid ]C[mid ] [ges ] 4, or G has an edge with one end in A and the other end in B, or one of A,B has at most one vertex. We show that any quasi 4-connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P−10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte's Four Flow Conjecture: every 2-edge-connected graph with no minor isomorphic to P−10 has a nowhere-zero 4-flow. This extends a result of Kilakos and Shepherd who proved the same for 3-regular graphs.

scholarly journals Colouring the Petals of a Graph

10.37236/1699 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
David Cariolaro ◽  
Gianfranco Cariolaro
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Petersen Graph ◽  
Empty Graph ◽  
Single Exception ◽  
Class 1

A petal graph is a connected graph $G$ with maximum degree three, minimum degree two, and such that the set of vertices of degree three induces a $2$–regular graph and the set of vertices of degree two induces an empty graph. We prove here that, with the single exception of the graph obtained from the Petersen graph by deleting one vertex, all petal graphs are Class $1$. This settles a particular case of a conjecture of Hilton and Zhao.

scholarly journals Long cycles in graphs with prescribed toughness and minimum degree

1995 ◽  
Vol 141 (1-3) ◽  
pp. 1-10 ◽  
Author(s):  
Douglas Bauer ◽  
H.J. Broersma ◽  
J. van den Heuvel ◽  
H.J. Veldman
Keyword(s):  
Minimum Degree ◽  
Long Cycles

On the Structure of Dense Triangle-Free Graphs

1999 ◽  
Vol 8 (3) ◽  
pp. 237-245 ◽  
Author(s):  
STEPHAN BRANDT
Keyword(s):  
Recent Result ◽  
Early Result ◽  
Infinite Sequence ◽  
Minimum Degree ◽  
Petersen Graph ◽  
Free Graph

As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a specific infinite sequence of graphs or contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that, if a (not necessarily maximal) triangle-free graph of order n has minimum degree δ[ges ]n/3, then the graph is either homomorphic with a member of the indicated family or contains the Petersen graph with one edge contracted. As a corollary we get a recent result due to Chen, Jin and Koh. Finally, we show that every triangle-free graph with δ>n/3 is either homomorphic with C5 or contains the Möbius ladder. A major tool is the observation that every triangle-free graph with δ[ges ]n/3 has a unique maximal triangle-free supergraph.

scholarly journals The Threshold Probability for Long Cycles

2016 ◽  
Vol 26 (2) ◽  
pp. 208-247 ◽  
Author(s):  
ROMAN GLEBOV ◽  
HUMBERTO NAVES ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Complete Graph ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Threshold Probability ◽  
Spanning Subgraph ◽  
Classic Result ◽  
Long Cycles

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

scholarly journals Large Feedback Arc Sets, High Minimum Degree Subgraphs, and Long Cycles in Eulerian Digraphs

2013 ◽  
Vol 22 (6) ◽  
pp. 859-873 ◽  
Author(s):  
HAO HUANG ◽  
JIE MA ◽  
ASAF SHAPIRA ◽  
BENNY SUDAKOV ◽  
RAPHAEL YUSTER
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Constant Factor ◽  
Feedback Arc Set ◽  
Eulerian Digraph ◽  
Long Cycles

A minimum feedback arc set of a directed graph G is a smallest set of arcs whose removal makes G acyclic. Its cardinality is denoted by β(G). We show that a simple Eulerian digraph with n vertices and m arcs has β(G) ≥ m2/2n2+m/2n, and this bound is optimal for infinitely many m, n. Using this result we prove that a simple Eulerian digraph contains a cycle of length at most 6n2/m, and has an Eulerian subgraph with minimum degree at least m2/24n3. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in Eulerian digraphs.

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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