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Description
Fractional Graph Theory: A Rational Approach to the Theory of GraphsA unified treatment of the most important results in the study of fractional graph concepts, this volume explores the various ways in which integer valued concepts can be modified to derive nonintegral values. It begins with the general fractional theory of hypergraphs and presents in depth coverage of fundamental and advanced topics. Subjects include fractional matching, fractional coloring, fractional edge coloring, fractional arboricity via matroid
A unified treatment of the most important results in the study of fractional graph concepts, this volume explores the various ways in which integer-valued concepts can be modified to derive nonintegral values. It begins with the general fractional theory of hypergraphs and presents in-depth coverage of fundamental and advanced topics. Subjects include fractional matching, fractional coloring, fractional edge coloring, fractional arboricity via matroid methods, and fractional isomorphism. The final chapter examines additional topics such as fractional domination, fractional intersection numbers, and fractional aspects of partially ordered sets.Challenging exercises reinforce the contents of each chapter, and the authors provide substantial references and bibliographic materials. A comprehensive reference for researchers, this volume also constitutes an excellent graduate-level text for students of graph theory and linear programming.
Binding Type: Paperback
Publisher: Dover Publications
Published: 11/16/2011
ISBN: 9780486485935
Pages: 211
Weight: 0.65lbs
Size: 8.90h x 5.90w x 0.60d
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★★★★★ 5
My dog loves it
My 70 lb 1 year old dog plays his own soccer with this toy and he loves making it squeek but he is a toy destroyer and so far this has held up pretty well. It's made of a soft material so I don't have to worry about it harming his teeth. It's easy to clean. It can be used in the house or outside for fetch. It is unpredictable at times with its bouncing which makes it more interesting for my dog to chase after. This is his favorite toy and carries it everywhere including to bed.
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Reviewed in the United States on September 23, 2025
★★★★★ 5
Durable Enough for Strong Jaws
I’ve gone through many toys that my dog destroys in minutes. This one has held up way better. The material feels tough, yet safe for chewing. I use it both indoors and outside, and it’s easy to rinse off when dirty. It’s been a relief to find something that can last longer in daily play.
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Reviewed in the United States on September 23, 2025
★★★★★ 5
Long lasting and safer alternative to sticks
Color: Dogwood & Calming, Size: Medium
These are great, very durable, and have lasted a very long time. Much safer for my dogs than the random sticks they find in the yard. They do get shorter with chewing eventually, but they don't break off in little chunks like I've seen with some nylon chews, and they don't splinter like wood. Medium size works well for both my 16 lb Cavalier and my 60 lb Pyrenees/Golden mix. They're both moderate-to-aggressive chewers, though size obviously affects how much damage they can do.
I wish you could buy the hemp chew separately because that one is very much a favorite in my house, so after 2 years and 2 purchases, I have barely any left of the remaining hemp chew, but still 2 of the dogwood chews that neither dog is as interested in anymore. I'd love to buy a couple more of just the hemp one.
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Reviewed in the United States on April 30, 2026
★★★★★ 5
Your dog will thank you.
Color: Dogwood & Fresh Breath, Size: Large
My dogs loves to chew on these. They make a small mess but not as bad as other chews. They are food for keeping teeth clean.
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Reviewed in the United States on May 31, 2026
★★★★★ 5
Nice chew toy
Color: Dogwood Mushroom, Size: Medium, Color: Dogwood Mushroom, Size: Medium
Super durable and puppy loves it. It was a bit hard for her at first but now at 5 months it’s one of her favorite things to gnaw on! It is heavy for the size but seems to be great quality and has given many hours of chew time with minimal wear.
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Reviewed in the United States on March 12, 2026
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