Linda Lesniak

Western Michigan University | WMU · Department of Mathematics

In 1972 Chvátal gave a well-known sufficient condition for a graphical sequence to be forcibly Hamiltonian, and showed that in some sense his condition is best possible. In this paper, we conjecture that with probability 1 as n → ∞ , Chvátal’s sufficient condition is also necessary. In contrast, we essentially prove that for every k ≥ 1 , the suffi...

Let ℓ be a positive integer, k = 2ℓ or k = 2ℓ + 1, and let n be a positive integer with n ≡ 1 (mod 2ℓ+1). For a prime p, n(p) denotes the largest integer i such that pi divides n. Potočnik and Šajna showed that if there exists a vertex-transitive self-complementary k-hypergraph of order n, then for every prime p we have pn(p) ≡ 1 (mod 2ℓ+1). Here w...

We show, with simple combinatorics, that if the dimples on a golf ball are all 5-sided and 6-sided polygons, with three dimples at each "vertex", then no matter how many dimples there are and no matter the sizes and distribution of the dimples, there will always be exactly twelve 5-sided dimples. Of course, the same is true of a soccer ball and its...

In [Discrete Math., 311 (2011), 688-689], Fujita defined f(r,n) to be the maximum integer κ such that every r-edge-coloring of Kn contains a monochromatic cycle of length at least κ. In this paper we investigate the values of f(r,n) when n is linear in r. We determine the value of f (r, 2r+2) for all r ≥ 1 and show that f (r, sr+c) - s+1 if r is su...

Let c, m, n be integers with n⩾c⩾3n⩾c⩾3, and let G be a graph of order n with m edges. A classical theorem by Erdős and Gallai states that if m>(c−1)(n−1)/2m>(c−1)(n−1)/2, then G contains a cycle of order at least c. In fact, the bound on m in this theorem is not sharp when both n and c are even. In this work we give the sharp value for this case.

Graphs & Digraphs masterfully employs student-friendly exposition, clear proofs, abundant examples, and numerous exercises to provide an essential understanding of the concepts, theorems, history, and applications of graph theory. Fully updated and thoughtfully reorganized to make reading and locating material easier for instructors and students, t...

Let G be a graph of order n and circumference c(G). Let G¯ be the complement of G. We prove that max{c(G),c(G¯)}≥⌈2n3⌉ and show sharpness of this bound.

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. In this paper, we look for obstruction sets beyond these sets. We introduce th...

Summary This project uses the geodesic dome made famous by Buckminster Fuller to extend students' understanding of Eulerian graphs. Students are directed to build a dome and, after determining that the graphical representation of the dome contains neither an Euler cycle nor trail, to find the minimal number of repeated edges that are necessary to v...

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. In this paper, we find this number for the alternating group graphs, Cayley graphs generated by 2-trees and the (n,k)-arrangement graphs. Moreover, we classify all the optimal so...

Let G be a connected graph and let c:V(G)→{1,2,⋯,k} be a coloring of the vertices of G for some positive integer k (where adjacent vertices may be colored the same). The color code of a vertex v of G (with respect to c) is the ordered (k+1)-tuple code (v)=(a 0 ,a 1 ,⋯,a k ) where a 0 is the color assigned to v and for 1≤i≤k, a i is the number of ve...

A variety of recent developments in hamiltonian theory are reviewed. In particular, several sufficient conditions for a graph to be hamiltonian, certain hamiltonian properties of line graphs, and various hamiltonian properties of powers of graphs are discussed. Furthermore, the concept of an n-distant hamiltonian graph is introduced and several the...

In this note, we consider a minimum degree condition for a hamiltonian graph to have a 2-factor with two components. Let G be a graph of order n⩾3. Dirac's theorem says that if the minimum degree of G is at least , then G has a hamiltonian cycle. Furthermore, Brandt et al. [J. Graph Theory 24 (1997) 165–173] proved that if n⩾8, then G has a 2-facto...

Given positive integers k m n, a graph G of order n is (k, m)-pancyclic ordered if for any set of k vertices of G and any integer r with m r n, there is a cycle of length r encountering the k vertices in a specified order. Minimum degree conditions that imply a graph of sufficiently large order n is (k, m)-pancylic ordered are proved. Examples show...

In this paper we generalize a Theorem of Jung which shows that 1-tough graphs with δ(G)⩾|V(G)|-42 are hamiltonian. Our generalization shows that these graphs contain a wide variety of 2-factors. In fact, these graphs contain not only 2-factors having just one cycle (the hamiltonian case) but 2-factors with k cycles, for any k such that 1⩽k⩽n-164.

Given positive integers kmn, a graph G of order n is (k,m)-pancyclic if for any set of k vertices of G and any integer r with mrn, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k,m)-pancylic are proved. If the additional property...

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of order n with 3 ≤ k ≤ n/2, and deg(u) + deg(v) ≥ n + (3k − 9)/2 for every pair u, v of nonadjacent vertices of G, th...

In this paper we characterize those forbidden triples of graphs, no one of which is a generalized claw, sufficient to imply that a 2-connected graph of sufficiently large order is hamiltonian.

In this paper we characterize those forbidden triples of graphs, no one of which is a generalized claw, sufficient to imply that a 2-connected graph of sufficiently large order is hamiltonian.

Ng and Schultz [J Graph Theory 1 (1997), 45–57] introduced the idea of cycle orderability. For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k-ordered Hamiltonia...

Ng and Schultz [J Graph Theory 1 (1997), 45–57] introduced the idea of cycle orderability. For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k-ordered Hamiltonia...

. In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor...

The girth of a graph with a Hamiltonian cycle and t chords is investigated. In particular, for any integer t>0 let g(t) denote the smallest number such that any Hamiltonian graph G with n vertices and n+t edges has girth at most g(t)n+c, where c is a constant independent of n. It is shown that there exist constants c 1 and c 2 such that (c 1 (logt)...

For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ore's classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We a...

For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ore's classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We a...

The k-spectrum sk(G) of a graph G is the set of all positive integers that occur as the size of an induced k-vertex subgraph of G. In this paper we determine the minimum order and size of a graph G with sk (G) = {0, 1, …,(2k)} and consider the more general question of describing those sets S ⊆ {0,1, … ,(2k)} such that S = sk(G) for some graph G.

The k-spectrum sk(G) of a graph G is the set of all positive integers that occur as the size of an induced k-vertex subgraph of G. In this paper we determine the minimum order and size of a graph G with sk (G) = {0, 1, …,(2k)} and consider the more general question of describing those sets S ⊆ {0,1, … ,(2k)} such that S = sk(G) for some graph G.

One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graphG has orderp and minimum degree at least thenG is hamiltonian. Moon and Moser showed that a balanced bipartite graph (the two partite sets have the same order)G has orderp and minimum degree more than thenG is hamiltonian. In this paper, their idea is...

A non-Hamiltonian cycle C in a graph G is extendable if there is a cycle C′ in G with V(C′) ⊃ V(C) with one more vertex than C. For any integer k ⩾ 0, a cycle C is k-chord extendable if it is extendable to the cycle C′ using at most k of the chords of the cycle C. It will be shown that if G is a graph of order n, then δ(G) > 3n/4 − 1 implies that a...

We consider a generalized degree condition based on the cardinality of the neighborhood union of arbitrary sets of r vertices. We show that a Dirac-type bound on this degree in conjunction with a bound on the independence number of a graph is sufficient to imply certain hamiltonian properties in graphs. For K1,m-free grphs we obtain generalizations...

A non-Hamiltonian cycle C in a graph G is extendable if there is a cycle C′ in G with V(C′) ⊃ V(C) with one more vertex than C. For any integer k ⩾ 0, a cycle C is k-chord extendable if it is extendable to the cycle C′ using at most k of the chords of the cycle C. It will be shown that if G is a graph of order n, then δ(G) > 3n/4 − 1 implies that a...

A caterpillar is a tree with the property that the vertices of degree at least 2 induce a path. We show that for every graph G of order n, either G or G ¯ has a spanning caterpillar of diameter at most 2logn. Furthermore, we show that if G is a graph of diameter 2 (diameter 3), then G contains a spanning caterpillar of diameter at most cn 3/4 (at m...

Let (x, y) be an edge of a graph G. Then the rotation of (x, y) about x is the operation of removing (x, y) from G and inserting (x, y′) as an edge, where y′ is a vertex of G. The rotation distance between graphs G and H is the minimum number of rotations necessary to transform G into H. Lower and upper bounds are given on the rotation distance of...

Let (x,y) be an edge of a graph G. Then the rotation of (x, y) about xis the operation of removing (x, y) from G and inserting (x, y') as an edge, where y' is a vertex of G. The rotation distance between graphs G and H is the minimum number of rotations necessary to transform G into H. Lower and upper bounds are given on the rotation distance of tw...

We prove that for d ≥ 4, d ≠ 5, the edges of the d-dimensional cube can be colored by d colors so that all quadrangles have four distinct colors. © 1993 John Wiley & Sons, Inc.

In several papers a variety of questions have been raised concerning the existence of cycles of length 0 mod k in graphs. For the case k=4, we answer three of these questions by showing that a graph G contains such a cycle provided it has any of the following three properties: (1) G has minimum degree at least 2 and at most two vertices of degree 2...

In this paper we introduce a class of graphs called ø-threshold graphs which generalize threshold graphs. These graphs are studied for various functions ø that include “max” and “min” as special cases. The graphs are then shown to fit into the more general setting of ø-tolerance intersection graphs previously introduced by Jacobson, McMorris, and M...

Dirac proved that if each vertex of a graph G of order n⩾3 has degree at least n/2, then the graph is Hamiltonian. This result will be generalized by showing that if the union of the neighborhoods of each pair of vertices of a 2-connected graph G of sufficiently large order n has at least n/2 vertices, then G is Hamiltonian. Other results that are...

For positive integers d and m, let Pd,m(G) denote the property that between each pair of vertices of the graph G, there are m internally disjoint paths of length at most d. For a positive integer t, a graph G satisfies the minimum generalized degree condition δt(G) ≥ s if the cardinality of the union of the neighborhoods of each set of t vertices o...

In this paper we use independent generalized degree conditions imposed on K(1, m)-free graphs (for an integer m⩾3) to obtain results involving β(G), the vertex independence number of G. We determine that in a K(1, m)-free graph G of order n if the cardinality of the neighborhood union of pairs of non-adjacent vertices is a positive fraction of n, t...

It is known that if a 2-connected graphG of sufficiently large ordern satisfies the property that the union of the neighborhoods of each pair of vertices has cardinality at leastn/2, thenG is hamiltonian. In this paper, we obtain a similar generalization of Dirac’s Theorem forK(1,3)-free graphs. In particular, we show that ifG is a 2-connectedK(1,3...

A graph G satisfies the neighborhood condition ANC(G) ⩾ m if, for all pairs of vertices of G, the union of their neighborhoods has at least m vertices. For a fixed positive integer k, let G be a graph of even order n which satisfies the following conditions: δ(G) ⩾ k + 1; 1(G) ⩾ k; and ANC(G) ⩾ n/2. It is shown that if n is sufficiently large then...

If S is a set of vertices of a graph G, then the neighborhood N(S) of S is defined to be the set of all vertices of G which are adjacent to at least one vertex of S. A survey is presented of recent results of the following form: Given a graph G, if |N(S)| is at least k for every subset S of V(G) of a specified type, then G has property P. We look,...

For sets of vertices, we consider a form of generalized degree based on neighborhood unions. Bounding this generalized degree from below, we obtain results about the connectivity of a graph as well as Dirac-type results about highly Hamiltonian properties in such graphs. In particular, we determine lower bounds on the generalized degree sufficient...

For any graph G, let i(G) and μ;(G) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers m and n, the lower Ramsey number s(m, n) is the largest integer p so that every graph of order p has i(G) ≤ m or μ;(G) ≤ n. In this paper we give several new lower bounds for s (m, n) as wel...

The notion of considering properties in graphs which meet the condition that for all independent pairs of vertices, x and y , deg( x ) + deg( y ) ≥ s , for some integer s , was first done by Ore. Recently, the concept of replacing degree sum by the order of the union of the neighborhoods has been considered. This was generalized to considering neig...

The closure C *(C) of a graph G with n vertices is obtained by iteratively adding edges to G between nonadjacent vertices whose degrees sum to at least n. We show that for almost all graphs G with edge probability at least 1/2, C *(G) is a complete graph, answering a question posed by Palmer.

The bipartite regulation number br(G) of a bipartite graph G with maximum degree d is the minimum number of vertices required to add to G to construct a d-regular bipartite supergraph of G. It is shown that if G is a connected m-by-n bipartite graph with m ⩽ n and n − m ⩾ d − 1, then br(G) = n − m. If. however, n − m ⩽ d − 2, the br(G) = n ⇔ m + 2l...

An overview of Eulerian graphs is presented. In particular, characterizations of Eulerian graphs and digraphs as well as algorithms for constructing Eulerian circuits are discussed. A solution to the Chinese postman problem is followed by a study of subgraphs and supergraphs of Eulerian graphs. After an introduction to randomly Eulerian graphs and...

A graph F is stable if for all n sufficiently large and all graphs G of order n, It is shown that if F is a graph which is not a star, then F is stable if and only if given any edge w1w2 of F there are nonadjacent vertices w3, w4 with F−{w3, w4} ⊂ F − {w1,w2}.

With each nonempty graph G one can associate a graph L(G), called the line graph of G, with the property that there exists a one-to-one correspondence between E(G) and V(L(G)) such that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. For integers m ≥ 2, the mth iterated line graph Lm(G) of G is defined to...

In graph theory, the related problems of deciding when a set of vertices or a set of edges constitutes a maximum matching or a minimum covering have been extensively studied. In this paper we generalize these ideas by defining total matchings and total coverings, and show that these sets, whose elements in general consist of both vertices and edges...

A set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E . If the subgraph 〈 E 〉 induced by E is a trail T , then T is called a dominating trail of G . Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and...

A graph G is said to have a factorization into the subgraphs G1,…, Gk if the subgraphs are spanning, pairwise edge-disjoint, and the union of their edge sets equals the edge set of G. For a graphical parameter f and positive integers n1, n2,…, nk (k ≧ 1), the f-Ramsey number f(n1, n2,…, nk) is the least positive integer p such that for any factoriz...

The degree setD D of a digraphD is the set of outdegrees of the vertices ofD. For a finite, nonempty setS of nonnegative integers, it is shown that there exists an asymmetric digraph (oriented graph)D such thatD D =S. Furthermore, the minimum order of such a digraphD is determined. Also, given two finite sequences of nonnegative integers, a necessa...

The eccentricitye(v) of a vertexv of a connected graphG is the maximum distance fromv among the vertices ofG. A nondecreasing sequencea 1,a 2, ...,a p of nonnegative integers is said to be an eccentric sequence if there exists a connected graphG of orderp whose vertices can be labelledv 1,v 2, ...,v p so thate(v i )=a i for alli. Several proper...

It is shown that if G is a graph of order p ≥ 2 such that deg u + deg v ≥ p − 1 for all pairs u, v of nonadjacent vertices, then the edge-connectivity of G equals the minimum degree of G. Furthermore, if deg u + deg v ≥ p for all pairs u, v of nonadjacent vertices, then either p is even and G is isomorphic to or every minimum cutset of edges of G c...

The cube G3 of a connected graph G is that graph having the same vertex set as G and in which two distinct vertices are adjacent if and only if their distance in G is at most three. A Hamiltonian-connected graph has the property that every two distinct vertices are joined by a Hamiltonian path. A graph G is 1-Hamiltonian-connected if, for every ver...

Let x1(G) denote the edge chromatic number of the graph G. For any positive integer m and any graph G, the mixed Ramsey number x1(m,G) is defined as the least positive integer p such that if H is any graph of order p, then either x1(H) ≥ m or [`(H)]\overline H contains a subgraph isomorphic to G. The number x1(m,G) is investigated for certain clas...