Reproduced with permission of the copyright owner. Further reproduction prohibited without permis... more Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s h i p freely. F i n a l l y , I w o u l d li k e t o t h a n k P r o f e s s o r E. A. N o r d h a u s of M i c h i g a n S t a t e U n i v e r s i t y a n d P r o f e s s o r C. E. W a l l of O l d D o m i n i o n U n i v e r s i t y for s e r v i n g o n m y c o m m i t t e e , a nd D i a n e C o r r a d i n i for t y p i n g t h i s m a n u s c r i p t . L i n d a M. L e s n i a k ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. re p e a t e d . e s u a n d v o f G , d e n o t e d <3q (u ,v ) , is t h e l e n g t h o f a s h o r t e s t u -v p a t h in G . T h e d i a m e t e r o f a c o n n e c t e d g r a p h G , d e n o t e d d i a m G , is d e f i n e d b y d i a m G = m a x d (u,v) . u , v € V ( G ) A n e u l e r i a n t r a i l o f a c o n n e c t e d g r a p h G is a n o p e n t r a i l o f G c o n t a i n i n g a l l the e d g e s o f G , w h i l e an e u l e r i a n c i r c u i t o f G is a c i r c u i t o f G c o n t a i n i n g all t h e e d g e s of G . A g r a p h p o s s e s s i n g an e u l e r i a n c i r c u i t is c a l l e d a n e u l e r i a n g r a p h . T h e f o l l o w i n g w e l l -k n o w n c h a r a c t e r i z a t i o n s of e u l e r i a n g r a p h s a n d g r a p h s w i t h e u l e r i a n t r a i l s are f r e q u e n t l y u s e d in C h a p t e r s IV a n d V. G is even. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s b o u n d c a n b e s u b s t a n t i a l l y improved. In o r d e r to p r e s e n t t h e s e theor e m s , w e f i rst p r o v e t h e f o l l o w i n g lemma. L e m m a 2-1. If G is a n o n -h a m i l t o n i a n g r a p h of o r d e r p ^ 3 w h i c h c o n t a i n s a h a m i l t o n i a n p a t h '^2 " " ' ^p ' t h e n d e g v^ + d e g v^ ^ p -1 . M o r e o v e r , if d e g v^ + d e g Vp = p -1 , t h e n t h e f o l l o w i n g m u s t hold: (i) if / E(G) , 2 :s k ^ p , t h e n (ii) if p is e v e n a n d d e g v^^ ^ deg v^ , t h e n and e d ges o f G f or s o m e I s a t i s f y i n g 2 :s t ^ p -2 . P r o o f . That deg v^ + deg v^ p -1 follows from a re sult in [20, p. 55]. Suppose deg v^ + deg v^ = p -1 . Vi=[Vi [ l:sisp-l a n d v^^v^^^^ 6 E ( G)} a n d V 2 = [Vi I 1 g i g p -1 a n d v^v^^^^ E(G)] . T h e n U ^2 / ..., v^_^ ) a n d H V g = (6 , i m p l y i n g t h a t JVj^l + [ V g l = p -1 . M o r e o v e r , | = d e g v^ . S i n c e deg v^ + d e g v^ = p -1 , w e c o n c l u d e t h a t d e g v^ = (Vg| . Now, v is a d j a c e n t to n o v e r t e x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
We consider the problem of finding long cycles in balanced tripartite graphs. We survey the relev... more We consider the problem of finding long cycles in balanced tripartite graphs. We survey the relevant literature, namely degree and edge conditions for Hamiltonicity and long cycles in graphs, including bipartite and k-partite results where they exist. We then prove that if G is a balanced tripartite graph on 3n vertices, G must contain a cycle of length at least 3n − 1, provided that e(G) ≥ 3n2 − 4n + 5 and n ≥ 14.
It is shown that if G is a graph of order p 2 2 such that deg u + deg u 2 p-1 for all pairs u, u ... more It is shown that if G is a graph of order p 2 2 such that deg u + deg u 2 p-1 for all pairs u, u of nonadjacent vertices, then the edge-connectivity of G equals the minimum degree of G. Furthermore, if deg u + deg u _> p for all pairs u, u of nonadjacent vertices, then either p is even and G is isomorphic to Kpj2 X K, or every minimum cutset of edges of G consists of the collection of edges incident with a vertex of least degree.
For any positive integer k and any (2 + k - n)-connected graph of order n, we define, following B... more For any positive integer k and any (2 + k - n)-connected graph of order n, we define, following Bondy and Chvatal, the k-neighborhood closure NC(k)(G) as the graph obtained from G by recursively joining pairs of nonadjacent vertices a, b satisfying the condition vertical bar N(a) boolean OR N(b)vertical bar + delta(ab) + epsilon(ab) >= k, where delta(ab) = min {d(x)vertical bar a, b is not an element of N(x) boolean OR {x}} and epsilon(ab) is a well defined binary variable. For many properties P of G, there exists a suitable k (depending on P and n) such that NC(k)(G) has property P if and only if G does. (c) 2008 Elsevier B.V. All rights reserved.
In 2009, Adamus showed that if $G$ is a balanced tripartite graph of order $3n$, $n \geq 2$, with... more In 2009, Adamus showed that if $G$ is a balanced tripartite graph of order $3n$, $n \geq 2$, with at least $3n^2 - 2n + 2$ edges, then $G$ is hamiltonian and, in fact, $G$ is pancyclic. Removing all but one edge incident with any vertex of the complete, balanced tripartite graph $K(n,n,n)$ shows that this result is best possible. Here we extend the result to balanced $k$-partite graphs of order $kn$. We prove that for all integers $k\geq 3$ and $n\geq 1$, every balanced $k$-partite graph with $kn$ vertices and at least ${{(k^2-k)n^2-2n(k-1)+4}\over 2}$ edges is pancyclic. We also prove a similar result for $k$-partite graphs that are not balanced.
Since graph theory was considered to have begun some 275 years ago, it has evolved into a subject... more Since graph theory was considered to have begun some 275 years ago, it has evolved into a subject with a fascinating history, a host of interesting problems and numerous diverse applications. While graph theory has developed ever-increasing connections with other areas of mathematics and a variety of scholarly fields, it is its beauty that has attracted so many to it. As with the previous editions, the objective of this fifth edition is to describe much of the story that is graph theory – in terms of its concepts, its theorems, its applications and its history. Here too, the audience for the fifth edition is beginning graduate students and advanced undergraduate students. The main prerequisite required of students using this book is a knowledge of mathematical proofs. Some elementary knowledge of linear algebra and group theory is also useful for some topics. Although a one-semester course in graph theory using this text can be designed by selecting topics of greatest interest to th...
Continuing to provide a carefully written, thorough introduction, Graphs & Digraphs, Fifth Ed... more Continuing to provide a carefully written, thorough introduction, Graphs & Digraphs, Fifth Edition expertly describes the concepts, theorems, history, and applications of graph theory. Nearly 50 percent longer than its bestselling predecessor, this edition reorganizes the material and presents many new topics. New to the Fifth Edition New or expanded coverage of graph minors, perfect graphs, chromatic polynomials, nowhere-zero flows, flows in networks, degree sequences, toughness, list colorings, and list edge colorings New examples, figures, and applications to illustrate concepts and theorems Expanded historical discussions of well-known mathematicians and problems More than 300 new exercises, along with hints and solutions to odd-numbered exercises at the back of the book Reorganization of sections into subsections to make the material easier to read Bolded definitions of terms, making them easier to locate Despite a field that has evolved over the years, this student-friendly, classroom-tested text remains the consummate introduction to graph theory. It explores the subjects fascinating history and presents a host of interesting problems and diverse applications.
We prove that for any integers p ≥ k ≥ 3 and any k-tuple of positive integers (n1,. .. , n k) suc... more We prove that for any integers p ≥ k ≥ 3 and any k-tuple of positive integers (n1,. .. , n k) such that p = k i=1 ni and n1 ≥ n2 ≥. .. ≥ n k , the condition n1 ≤ p 2 is necessary and sufficient for every subgraph of the complete k-partite graph K(n1,. .. , n k) with at least 4 − 2p + 2n1 + k i=1 ni(p − ni) 2 edges to be chorded pancyclic. Removing all but one edge incident with any vertex of minimum degree in K(n1,. .. , n k) shows that this result is best possible. Our result implies that for any integers, k ≥ 3 and n ≥ 1, a balanced k-partite graph of order kn with has at least (k 2 −k)n 2 −2n(k−1)+4 2 edges is chorded pancyclic. In the case k = 3, this result strengthens a previous one by Adamus, who in 2009 showed that a balanced tripartite graph of order 3n, n ≥ 2, with at least 3n 2 − 2n + 2 edges is pancyclic.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permis... more Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s h i p freely. F i n a l l y , I w o u l d li k e t o t h a n k P r o f e s s o r E. A. N o r d h a u s of M i c h i g a n S t a t e U n i v e r s i t y a n d P r o f e s s o r C. E. W a l l of O l d D o m i n i o n U n i v e r s i t y for s e r v i n g o n m y c o m m i t t e e , a nd D i a n e C o r r a d i n i for t y p i n g t h i s m a n u s c r i p t . L i n d a M. L e s n i a k ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. re p e a t e d . e s u a n d v o f G , d e n o t e d <3q (u ,v ) , is t h e l e n g t h o f a s h o r t e s t u -v p a t h in G . T h e d i a m e t e r o f a c o n n e c t e d g r a p h G , d e n o t e d d i a m G , is d e f i n e d b y d i a m G = m a x d (u,v) . u , v € V ( G ) A n e u l e r i a n t r a i l o f a c o n n e c t e d g r a p h G is a n o p e n t r a i l o f G c o n t a i n i n g a l l the e d g e s o f G , w h i l e an e u l e r i a n c i r c u i t o f G is a c i r c u i t o f G c o n t a i n i n g all t h e e d g e s of G . A g r a p h p o s s e s s i n g an e u l e r i a n c i r c u i t is c a l l e d a n e u l e r i a n g r a p h . T h e f o l l o w i n g w e l l -k n o w n c h a r a c t e r i z a t i o n s of e u l e r i a n g r a p h s a n d g r a p h s w i t h e u l e r i a n t r a i l s are f r e q u e n t l y u s e d in C h a p t e r s IV a n d V. G is even. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s b o u n d c a n b e s u b s t a n t i a l l y improved. In o r d e r to p r e s e n t t h e s e theor e m s , w e f i rst p r o v e t h e f o l l o w i n g lemma. L e m m a 2-1. If G is a n o n -h a m i l t o n i a n g r a p h of o r d e r p ^ 3 w h i c h c o n t a i n s a h a m i l t o n i a n p a t h '^2 " " ' ^p ' t h e n d e g v^ + d e g v^ ^ p -1 . M o r e o v e r , if d e g v^ + d e g Vp = p -1 , t h e n t h e f o l l o w i n g m u s t hold: (i) if / E(G) , 2 :s k ^ p , t h e n (ii) if p is e v e n a n d d e g v^^ ^ deg v^ , t h e n and e d ges o f G f or s o m e I s a t i s f y i n g 2 :s t ^ p -2 . P r o o f . That deg v^ + deg v^ p -1 follows from a re sult in [20, p. 55]. Suppose deg v^ + deg v^ = p -1 . Vi=[Vi [ l:sisp-l a n d v^^v^^^^ 6 E ( G)} a n d V 2 = [Vi I 1 g i g p -1 a n d v^v^^^^ E(G)] . T h e n U ^2 / ..., v^_^ ) a n d H V g = (6 , i m p l y i n g t h a t JVj^l + [ V g l = p -1 . M o r e o v e r , | = d e g v^ . S i n c e deg v^ + d e g v^ = p -1 , w e c o n c l u d e t h a t d e g v^ = (Vg| . Now, v is a d j a c e n t to n o v e r t e x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
We consider the problem of finding long cycles in balanced tripartite graphs. We survey the relev... more We consider the problem of finding long cycles in balanced tripartite graphs. We survey the relevant literature, namely degree and edge conditions for Hamiltonicity and long cycles in graphs, including bipartite and k-partite results where they exist. We then prove that if G is a balanced tripartite graph on 3n vertices, G must contain a cycle of length at least 3n − 1, provided that e(G) ≥ 3n2 − 4n + 5 and n ≥ 14.
It is shown that if G is a graph of order p 2 2 such that deg u + deg u 2 p-1 for all pairs u, u ... more It is shown that if G is a graph of order p 2 2 such that deg u + deg u 2 p-1 for all pairs u, u of nonadjacent vertices, then the edge-connectivity of G equals the minimum degree of G. Furthermore, if deg u + deg u _> p for all pairs u, u of nonadjacent vertices, then either p is even and G is isomorphic to Kpj2 X K, or every minimum cutset of edges of G consists of the collection of edges incident with a vertex of least degree.
For any positive integer k and any (2 + k - n)-connected graph of order n, we define, following B... more For any positive integer k and any (2 + k - n)-connected graph of order n, we define, following Bondy and Chvatal, the k-neighborhood closure NC(k)(G) as the graph obtained from G by recursively joining pairs of nonadjacent vertices a, b satisfying the condition vertical bar N(a) boolean OR N(b)vertical bar + delta(ab) + epsilon(ab) >= k, where delta(ab) = min {d(x)vertical bar a, b is not an element of N(x) boolean OR {x}} and epsilon(ab) is a well defined binary variable. For many properties P of G, there exists a suitable k (depending on P and n) such that NC(k)(G) has property P if and only if G does. (c) 2008 Elsevier B.V. All rights reserved.
In 2009, Adamus showed that if $G$ is a balanced tripartite graph of order $3n$, $n \geq 2$, with... more In 2009, Adamus showed that if $G$ is a balanced tripartite graph of order $3n$, $n \geq 2$, with at least $3n^2 - 2n + 2$ edges, then $G$ is hamiltonian and, in fact, $G$ is pancyclic. Removing all but one edge incident with any vertex of the complete, balanced tripartite graph $K(n,n,n)$ shows that this result is best possible. Here we extend the result to balanced $k$-partite graphs of order $kn$. We prove that for all integers $k\geq 3$ and $n\geq 1$, every balanced $k$-partite graph with $kn$ vertices and at least ${{(k^2-k)n^2-2n(k-1)+4}\over 2}$ edges is pancyclic. We also prove a similar result for $k$-partite graphs that are not balanced.
Since graph theory was considered to have begun some 275 years ago, it has evolved into a subject... more Since graph theory was considered to have begun some 275 years ago, it has evolved into a subject with a fascinating history, a host of interesting problems and numerous diverse applications. While graph theory has developed ever-increasing connections with other areas of mathematics and a variety of scholarly fields, it is its beauty that has attracted so many to it. As with the previous editions, the objective of this fifth edition is to describe much of the story that is graph theory – in terms of its concepts, its theorems, its applications and its history. Here too, the audience for the fifth edition is beginning graduate students and advanced undergraduate students. The main prerequisite required of students using this book is a knowledge of mathematical proofs. Some elementary knowledge of linear algebra and group theory is also useful for some topics. Although a one-semester course in graph theory using this text can be designed by selecting topics of greatest interest to th...
Continuing to provide a carefully written, thorough introduction, Graphs & Digraphs, Fifth Ed... more Continuing to provide a carefully written, thorough introduction, Graphs & Digraphs, Fifth Edition expertly describes the concepts, theorems, history, and applications of graph theory. Nearly 50 percent longer than its bestselling predecessor, this edition reorganizes the material and presents many new topics. New to the Fifth Edition New or expanded coverage of graph minors, perfect graphs, chromatic polynomials, nowhere-zero flows, flows in networks, degree sequences, toughness, list colorings, and list edge colorings New examples, figures, and applications to illustrate concepts and theorems Expanded historical discussions of well-known mathematicians and problems More than 300 new exercises, along with hints and solutions to odd-numbered exercises at the back of the book Reorganization of sections into subsections to make the material easier to read Bolded definitions of terms, making them easier to locate Despite a field that has evolved over the years, this student-friendly, classroom-tested text remains the consummate introduction to graph theory. It explores the subjects fascinating history and presents a host of interesting problems and diverse applications.
We prove that for any integers p ≥ k ≥ 3 and any k-tuple of positive integers (n1,. .. , n k) suc... more We prove that for any integers p ≥ k ≥ 3 and any k-tuple of positive integers (n1,. .. , n k) such that p = k i=1 ni and n1 ≥ n2 ≥. .. ≥ n k , the condition n1 ≤ p 2 is necessary and sufficient for every subgraph of the complete k-partite graph K(n1,. .. , n k) with at least 4 − 2p + 2n1 + k i=1 ni(p − ni) 2 edges to be chorded pancyclic. Removing all but one edge incident with any vertex of minimum degree in K(n1,. .. , n k) shows that this result is best possible. Our result implies that for any integers, k ≥ 3 and n ≥ 1, a balanced k-partite graph of order kn with has at least (k 2 −k)n 2 −2n(k−1)+4 2 edges is chorded pancyclic. In the case k = 3, this result strengthens a previous one by Adamus, who in 2009 showed that a balanced tripartite graph of order 3n, n ≥ 2, with at least 3n 2 − 2n + 2 edges is pancyclic.
Uploads
Papers by Linda Lesniak