31 research outputs found
A study on Dicycles and Eulerian Subdigraphs
1. Dicycle cover of Hamiltonian oriented graphs. A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs, including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.;2. Supereulerian digraphs with given local structures . Catlin in 1988 indicated that there exist graph families F such that if every edge e in a graph G lies in a subgraph He of G isomorphic to a member in F, then G is supereulerian. In particular, if every edge of a connected graph G lies in a 3-cycle, then G is supereulerian. The purpose of this research is to investigate how Catlin\u27s theorem can be extended to digraphs. A strong digraph D is supereulerian if D contains a spanning eulerian subdigraph. We show that there exists an infinite family of non-supereulerian strong digraphs each arc of which lies in a directed 3-cycle. We also show that there exist digraph families H such that a strong digraph D is supereulerian if every arc a of D lies in a subdigraph Ha isomorphic to a member of H. A digraph D is symmetric if (x, y) ∈ A( D) implies (y, x) ∈ A( D); and is symmetrically connected if every pair of vertices of D are joined by a symmetric dipath. A digraph D is partially symmetric if the digraph obtained from D by contracting all symmetrically connected components is symmetrically connected. It is known that a partially symmetric digraph may not be symmetrically connected. We show that symmetrically connected digraphs and partially symmetric digraphs are such families. Sharpness of these results are discussed.;3. On a class of supereulerian digraphs. The 2-sum of two digraphs D1 and D2, denoted D1 ⊕2 D2, is the digraph obtained from the disjoint union of D 1 and D2 by identifying an arc in D1 with an arc in D2. A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on D1 and D 2 for D1 ⊕2 D 2 to be supereulerian. In particular, we show that if D 1 and D2 are symmetrically connected or partially symmetric, then D1 ⊕2 D2 is supereulerian
Eulerian subgraphs and Hamiltonicity of claw -free graphs
Let C(l, k) denote the class of 2-edge-connected graphs of order n such that a graph G ∈ C(l, k) if and only if for every edge cut S ⊆ E(G) with |S| ≤ 3, each component of G - S has order at least n-kl . We prove that If G ∈ C(6, 0), then G is supereulerian if and only if G cannot be contracted to K2,3, K 2,5 or K2,3(e), where e ∈ E(K2,3) and K2,3(e) stands for a graph obtained from K2,3 by replacing e by a path of length 2. Previous results by Catlin and Li, and by Broersma and Xiong are extended.;We also investigate the supereulerian graph problems within planar graphs, and we prove that if a 2-edge-connected planar graph G is at most three edges short of having two edge-disjoint spanning trees, then G is supereulerian except a few classes of graphs. This is applied to show the existence of spanning Eulerian subgraphs in planar graphs with small edge cut conditions. We determine several extremal bounds for planar graphs to be supereulerian.;Kuipers and Veldman conjectured that any 3-connected claw-free graph with order n and minimum degree delta ≥ n+610 is Hamiltonian for n sufficiently large. We prove that if H is a 3-connected claw-free graph with sufficiently large order n, and if delta(H) ≥ n+510 , then either H is hamiltonian, or delta( H) = n+510 and the Ryjac˘ek\u27s closure cl( H) of H is the line graph of a graph obtained from the Petersen graph P10 by adding n-1510 pendant edges at each vertex of P10
Circuits and Cycles in Graphs and Matroids
This dissertation mainly focuses on characterizing cycles and circuits in graphs, line graphs and matroids. We obtain the following advances.
1. Results in graphs and line graphs. For a connected graph G not isomorphic to a path, a cycle or a K1,3, let pc(G) denote the smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and t, respectively, then P is called a divalent (s, t)-path. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K3}. We prove the following. (i) If G is a connected triangular graph, then L(G) is panconnected if and only if G is essentially 3-edge-connected. (ii) pc(G) ≤ l(G) + 2. Furthermore, if l(G) ≥ 2, then pc(G) = l(G) + 2 if and only if for some integer t ≥ 3, G has a bridge divalent (3, t)-path of length l(G).
For a graph G, the supereulerian width μ′(G) of a graph G is the largest integer s such
that G has a spanning (k;u,v)-trail-system, for any integer k with 1 ≤ k ≤ s, and for any
u, v ∈ V (G) with u ̸= v. Thus μ′(G) ≥ 2 implies that G is supereulerian, and so graphs with
higher supereulerian width are natural generalizations of supereulerian graphs. Settling an open
problem of Bauer, Catlin in [J. Graph Theory 12 (1988), 29-45] proved that if a simple graph
G on n ≥ 17 vertices satisfy δ(G) ≥ n − 1, then μ′(G) ≥ 2. In this paper, we show that for 4
any real numbers a, b with 0 \u3c a \u3c 1 and any integer s \u3e 0, there exists a finite graph family
F = F(a,b,s) such that for a simple graph G with n = |V(G)|, if for any u,v ∈ V(G) with
uv ∈/ E(G), max{dG(u), dG(v)} ≥ an + b, then either μ′(G) ≥ s + 1 or G is contractible to a
member in F. When a = 1,b = −3, we show that if n is sufficiently large, K3,3 is the only 42
obstacle for a 3-edge-connected graph G to satisfy μ′(G) ≥ 3. An hourglass is a graph obtained from K5 by deleting the edges in a cycle of length 4, and an
hourglass-free graph is one that has no induced subgraph isomorphic to an hourglass. Kriesell in [J. Combin. Theory Ser. B, 82 (2001), 306-315] proved that every 4-connected hourglass-free line graph is Hamilton-connected, and Kaiser, Ryj ́aˇcek and Vr ́ana in [Discrete Mathematics, 321 (2014) 1-11] extended it by showing that every 4-connected hourglass-free line graph is 1- Hamilton-connected. We characterize all essentially 4-edge-connected graphs whose line graph is hourglass-free. Consequently we prove that for any integer s and for any hourglass-free line
graph L(G), each of the following holds. (i) If s ≥ 2, then L(G) is s-hamiltonian if and only if κ(L(G)) ≥ s + 2; (ii) If s ≥ 1, then L(G) is s-Hamilton-connected if and only if κ(L(G)) ≥ s + 3.
For integers s1, s2, s3 \u3e 0, let Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, Ps3+1 of length s1,s2 and s3, respectively. We prove the following results. (i)LetN1 ={Ns1,s2,s3 :s1 \u3e0,s1 ≥s2 ≥s3 ≥0ands1+s2+s3 ≤6}. Thenforany N ∈ N1, every N-free line graph L(G) with |V (L(G))| ≥ s + 3 is s-hamiltonian if and only if κ(L(G)) ≥ s + 2. (ii)LetN2={Ns1,s2,s3 :s1\u3e0,s1≥s2≥s3≥0ands1+s2+s3≤4}.ThenforanyN∈N2, every N -free line graph L(G) with |V (L(G))| ≥ s + 3 is s-Hamilton-connected if and only if κ(L(G)) ≥ s + 3. 2. Results in matroids. A matroid M with a distinguished element e0 ∈ E(M) is a rooted matroid with e0 being the root. We present a characterization of all connected binary rooted matroids whose root lies in at most three circuits, and a characterization of all connected binary rooted matroids whose root lies in all but at most three circuits. While there exist infinitely many such matroids, the number of serial reductions of such matroids is finite. In particular, we find two finite families of binary matroids M1 and M2 and prove the following. (i) For some e0 ∈ E(M), M has at most three circuits containing e0 if and only if the serial reduction of M is isomorphic to a member in M1. (ii) If for some e0 ∈ E(M), M has at most three circuits not containing e0 if and only if the serial reduction of M is isomorphic to a member in M2. These characterizations will be applied to show that every connected binary matroid M with at least four circuits has a 1-hamiltonian circuit graph
On Eulerian subgraphs and hamiltonian line graphs
A graph {\color{black}} is Hamilton-connected if for any pair of distinct vertices {\color{black}}, {\color{black}} has a spanning -path; {\color{black}} is 1-hamiltonian if for any vertex subset with , has a spanning cycle. Let , and denote the minimum degree, the matching number and the line graph of a graph , respectively. The following result is obtained. {\color{black} Let be a simple graph} with . If , then each of the following holds. \\ (i) is Hamilton-connected if and only if . \\ (ii) is 1-hamiltonian if and only if . %==========sp For a graph , an integer and distinct vertices , an -path-system of is a subgraph consisting of internally disjoint -paths. The spanning connectivity is the largest integer such that for any with and for any with , has a spanning -path-system. It is known that , and determining if is an NP-complete problem. A graph is maximally spanning connected if . Let and be the smallest integers and such that is maximally spanning connected and , respectively. We show that every locally-connected line graph with connectivity at least 3 is maximally spanning connected, and that the spanning connectivity of a locally-connected line graph can be polynomially determined. As applications, we also determined best possible upper bounds for and , and characterized the extremal graphs reaching the upper bounds. %==============st For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is -supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in [J. Graph Theory, 12 (1988) 29-45] showed that every simple graph on vertices with , when is sufficiently large, is -supereulerian or is contractible to . We prove the following for any nonnegative integers and . \\ (i) For any real numbers and with , there exists a family of finitely many graphs \F(a,b;s,t) such that if is a simple graph on vertices with and , then either is -supereulerian, or is contractible to a member in \F(a,b;s,t). \\ (ii) Let denote the connected loopless graph with two vertices and parallel edges. If is a simple graph on vertices with and , then when is sufficiently large, either is -supereulerian, or for some integer with , is contractible to a . %==================index For a hamiltonian property \cp, Clark and Wormold introduced the problem of investigating the value \cp(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: and , and proposed a few problems to determine \cp(a,b) with when \cp is being hamiltonian, edge-hamiltonian and hamiltonian-connected. Zhan in 1986 proved that the line graph of a 4-edge-connected graph is Hamilton-connected, which implies a solution to the unsettled cases of above-mentioned problem. We consider an extended version of the problem. Let denote the essential edge-connectivity of a graph , and define \cp\u27(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: and . We investigate the values of \cp\u27(a,b) when \cp is one of these hamiltonian properties. In particular, we show that for any values of , \cp\u27(4,b) \le 2 and \cp\u27(4,b) = 1 if and only if Thomassen\u27s conjecture that every 4-connected line graph is hamiltonian is valid
On Generalizations of Supereulerian Graphs
A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let and be the edge-connectivity and the minimum degree of a graph , respectively. For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . This dissertation is devoted to providing some results on -supereulerian graphs and supereulerian hypergraphs.
In Chapter 2, we determine the value of the smallest integer such that every -edge-connected graph is -supereulerian as follows:
j(s,t) = \left\{ \begin{array}{ll} \max\{4, t + 2\} & \mbox{ if $0 \le s \le 1$, or $(s,t) \in \{(2,0), (2,1), (3,0),(4,0)\}$,} \\ 5 & \mbox{ if $(s,t) \in \{(2,2), (3,1)\}$,} \\ s + t + \frac{1 - (-1)^s}{2} & \mbox{ if $s \ge 2$ and $s+t \ge 5$. } \end{array} \right.
As applications, we characterize -supereulerian graphs when in terms of edge-connectivities, and show that when , -supereulerianicity is polynomially determinable.
In Chapter 3, for a subset with , a necessary and sufficient condition for to be a contractible configuration for supereulerianicity is obtained. We also characterize the -supereulerianicity of when . These results are applied to show that if is -supereulerian with , then for any permutation on the vertex set , the permutation graph is -supereulerian if and only if .
For a non-negative integer , a graph is -Hamiltonian if the removal of any vertices results in a Hamiltonian graph. Let and denote the smallest integer such that the iterated line graph is -supereulerian and -Hamiltonian, respectively. In Chapter 4, for a simple graph , we establish upper bounds for and . Specifically, the upper bound for the -Hamiltonian index sharpens the result obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].
Harary and Nash-Williams in 1968 proved that the line graph of a graph is Hamiltonian if and only if has a dominating closed trail, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint spanning trees is a contractible configuration for supereulerianicity. In Chapter 5, utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Kir\\u27aly and Kriesell in 2003, we generalize the above-mentioned results of Harary and Nash-Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are Hamiltonian, and showing that every 2-partition-connected hypergraph is a contractible configuration for supereulerianicity.
Applying the adjacency matrix of a hypergraph defined by Rodr\\u27iguez in 2002, let be the second largest adjacency eigenvalue of . In Chapter 6, we prove that for an integer and a -uniform hypergraph of order with even, the minimum degree and , if , then is -edge-connected. %.
Some discussions are displayed in the last chapter. We extend the well-known Thomassen Conjecture that every 4-connected line graph is Hamiltonian to hypergraphs. The -supereulerianicity of hypergraphs is another interesting topic to be investigated in the future
Small cycle cover, group coloring with related problems
Bondy conjectured that if G is a simple 2-connected graph with n ≥ 3 vertices, then the edges of G can be covered by at most 2n-33 cycles. In Chapter 2, a result on small cycle cover is obtained and we also show that the result is as best as possible.;Thomassen conjectured that every 4-connected line graph is hamiltonian. In Chapters 3 and 4, we apply Catlin\u27s reduction method to study cycles in line graphs. Results about hamiltonian connectivity of line graphs and 3-edge-connected graphs are obtained. Several former results are extended.;Jaeger, Linial, Payan and Tarsi introduced group coloring in 1992 and proved that the group chromatic number for every planar graph is at most 6. It is shown that the bound 6 can be decreased to 5. Jaeger, Linial, Payan and Tarsi also proved that the group chromatic number for every planar graph with girth at least 4 is at most 4. Chapters 5 and 6 are devoted to the study of group coloring of graphs.;The concept of list coloring, choosability and choice number was introduced by Erdos, Rubin and Taylor in 1979 and Vizing in 1976. Alon and Tarsi proved that every bipartite planar graph is 3-choosable. Thomassen showed that every planar graph is 5-choosable and that every planar graph with girth at least 5 is 3-choosable. In Chapter 7, results on list coloring are obtained, extending a former result of Thomassen
A study on supereulerian digraphs and spanning trails in digraphs
A strong digraph D is eulerian if for any v ∈ V (D), d+D (v) = d−D (v). A digraph D is supereulerian if D contains a spanning eulerian subdigraph, or equivalently, a spanning closed directed trail. A digraph D is trailable if D has a spanning directed trail. This dissertation focuses on a study of trailable digraphs and supereulerian digraphs from the following aspects.
1. Strong Trail-Connected, Supereulerian and Trailable Digraphs. For a digraph D, D is trailable digraph if D has a spanning trail. A digraph D is strongly trail- connected if for any two vertices u and v of D, D posses both a spanning (u, v)-trail and a spanning (v,u)-trail. As the case when u = v is possible, every strongly trail-connected digraph is also su- pereulerian. Let D be a digraph. Let S(D) = {e ∈ A(D) : e is symmetric in D}. A digraph D is symmetric if A(D) = S(D). The symmetric core of D, denoted by J(D), has vertex set V (D) and arc set S(D). We have found a well-characterized digraph family D each of whose members does not have a spanning trail with its underlying graph spanned by a K2,n−2 such that for any strong digraph D with its matching number α′(D) and arc-strong-connectivity λ(D), if n = |V (D)| ≥ 3 and λ(D) ≥ α′(D) − 1, then each of the following holds. (i) There exists a family D of well-characterized digraphs such that for any digraph D with α′(D) ≤ 2, D has a spanning trial if and only if D is not a member in D. (ii) If α′(D) ≥ 3, then D has a spanning trail. (iii) If α′(D) ≥ 3 and n ≥ 2α′(D) + 3, then D is supereulerian. (iv) If λ(D) ≥ α′(D) ≥ 4 and n ≥ 2α′(D) + 3, then for any pair of vertices u and v of D, D contains a spanning (u, v)-trail.
2. Supereulerian Digraph Strong Products. A cycle vertex cover of a digraph D is a collection of directed cycles in D such that every vertex in D lies in at least one dicycle in this collection, and such that the union of the arc sets of these directed cycles induce a connected subdigraph of D. A subdigraph F of a digraph D is a circulation if for every vertex v in F, the indegree of v equals its outdegree, and a spanning circulation if F is a cycle factor. Define f(D) to be the smallest cardinality of a cycle vertex cover of the digraph D/F obtained from D by contracting all arcs in F , among all circulations F of D. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if D1 and D2 are nontrivial strong digraphs such that D1 is supereulerian and D2 has a cycle vertex cover C′ with |C′| ≤ |V (D1)|, then the Cartesian product D1 and D2 is also supereulerian. We prove that for strong digraphs D1 and D2, if for some cycle factor F1 of D1, the digraph formed from D1 by contracting arcs in F1 is hamiltonian with f(D2) not bigger than |V (D1)|, then the strong product D1 and D2 is supereulerian