Journal ArticleDOI
Independence and graph homomorphisms
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A graph with n vertices that contains no triangle and no 5-cycle and minimum degree exceeding n/4 contains an independent set with at least (3n)/7 vertices, which is best possible.Abstract:
A graph with n vertices that contains no triangle and no 5-cycle and minimum degree exceeding n/4 contains an independent set with at least (3n)/7 vertices. This is best possible. The proof proceeds by producing a homomorphism to the 7-cycle and invoking the No Homomorphism Lemma. For k ≥ 4, a graph with n vertices, odd girth 2k+1, and minimum degree exceeding n/(k+1) contains an independent set with at least kn/(2k+1) vertices; however, we suspect this is not best possible. © 1993 John Wiley & Sons, Inc.read more
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Independence in Direct-Product Graphs.
Pranava K. Jha,Sandi Klavzar +1 more
TL;DR: It is shown that for any graph G with at least one edge and for any i ∈ IN there is a graph H such that α(G×H) >α(G–H) + i, and that this equality holds when G is a bipartite graph with a perfect matching and H is a traceable graph.
Journal ArticleDOI
On well-covered graphs of odd girth 7 or greater
TL;DR: It is proved that every connected member G of Gγ=α containing neither C3 nor C5 as a subgraph is a K1, C4, C7 or a corona graph.
Journal ArticleDOI
Circulants and Sequences
TL;DR: Zhou has shown that circulants and finite abelian Cayley graphs are stable and conjecture that $\mu(G)=\mu(S)$ for a reversible circulant with sufficiently many vertices, which agrees with the lower bound of Haggkvist for k=2 and of Albertson, Chan, and Haas for k 3.
Journal ArticleDOI
On the Structure of Graphs with Given Odd Girth and Large Minimum Degree
Silvia Messuti,Mathias Schacht +1 more
TL;DR: It is shown that every n-vertex graph with odd girth 2k-1 and minimum degree bigger than 34kn is homomorphic to the cycle of length 2k+1.
Journal ArticleDOI
Graphs of odd girth 7 with large degree
TL;DR: It is shown that every graph with minimum degree δ > 4 n / 17 and no odd cycles of length 3 or 5 is homomorphic with the Mobius ladder with 6 rungs and include the extremal graph characterization in the case of equality.
References
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Journal ArticleDOI
On the connection between chromatic number, maximal clique and minimal degree of a graph
TL;DR: A graph of n vertices, having chromatic number r which contains no complete graph of r Vertices, contains a vertex of degree not exceeding n(3r-7)/(3 r-4).
Journal ArticleDOI
A note on the star chromatic number
J. A. Bondy,Pavol Hell +1 more
TL;DR: A.A. Vince introduced a natural generalization of graph coloring and proved some basic facts, revealing it to be a concept of interest, and his work relies on continuous methods.
Journal ArticleDOI
Homomorphisms of 3-chromatic graphs
TL;DR: The principal result (Theorem 2) provides necessary conditions for the existence of a homomorphism onto a prescribed target and it is shown that iterated cartesian products of the Petersen graph form an infinite family of vertex transitive graphs no one of which is the homomorphic image of any other.