Edge-disjoint spanning trees and eigenvalues of regular graphs

Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of $k$ edge-disjoint spanning trees in a regular graph, when $k\in \{2,3\}$. More precisely, we show that if the second largest eigenvalue of a $d$-regular graph $G$ is less than $d-\frac{2k-1}{d+1}$, then $G$ contains at least $k$ edge-disjoint spanning trees, when $k\in \{2,3\}$. We construct examples of graphs that show our bounds are essentially best possible. We conjecture that the above statement is true for any $k<d/2$.Comment: 4 figure

Characterization of removable elements with respect to having k disjoint bases in a matroid

AbstractThe well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with k edge-disjoint spanning trees. Edmonds generalizes this theorem to matroids with k disjoint bases. For any graph G that may not have k-edge-disjoint spanning trees, the problem of determining what edges should be added to G so that the resulting graph has k edge-disjoint spanning trees has been studied by Haas (2002) [11] and Liu et al. (2009) [17], among others. This paper aims to determine, for a matroid M that has k disjoint bases, the set Ek(M) of elements in M such that for any e∈Ek(M), M−e also has k disjoint bases. Using the matroid strength defined by Catlin et al. (1992) [4], we present a characterization of Ek(M) in terms of the strength of M. Consequently, this yields a characterization of edge sets Ek(G) in a graph G with at least k edge-disjoint spanning trees such that ∀e∈Ek(G), G−e also has k edge-disjoint spanning trees. Polynomial algorithms are also discussed for identifying the set Ek(M) in a matroid M, or the edge subset Ek(G) for a connected graph G

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V ( G ) , G has a spanning ( u , v ) -trail. A graph G is edge-Eulerian-connected if for any e ′ and e ″ in E ( G ) , G has a spanning ( e ′ , e ″ ) -trail. For an integer r ⩾ 0 , a graph is called r -Eulerian-connected if for any X ⊆ E ( G ) with | X | ⩽ r , and for any u , v ∈ V ( G ) , G has a spanning ( u , v ) -trail T such that X ⊆ E ( T ) . The r -edge-Eulerian-connectivity of a graph can be defined similarly. Let θ ( r ) be the minimum value of k such that every k -edge-connected graph is r -Eulerian-connected. Catlin proved that θ ( 0 ) = 4 . We shall show that θ ( r ) = 4 for 0 ⩽ r ⩽ 2 , and θ ( r ) = r + 1 for r ⩾ 3 . Results on r -edge-Eulerian connectivity are also discussed

Completely Independent Spanning Trees in Line Graphs

Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this paper, we present a tight lower bound on the maximum number of completely independent spanning trees in $L(G)$, where $L(G)$ denotes the line graph of a graph $G$. Based on a new characterization of a graph with $k$ completely independent spanning trees, we also show that for any complete graph $K_n$ of order $n \geq 4$, there are $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees in $L(K_n)$ where the number $\lfloor \frac{n+1}{2} \rfloor$ is optimal, such that $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees still exist in the graph obtained from $L(K_n)$ by deleting any vertex (respectively, any induced path of order at most $\frac{n}{2}$) for $n = 4$ or odd $n \geq 5$ (respectively, even $n \geq 6$). Concerning the connectivity and the number of completely independent spanning trees, we moreover show the following, where $\delta(G)$ denotes the minimum degree of $G$. $\$ $\bullet$ Every $2k$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is not super edge-connected or $\delta(G) \geq 2k$. $\$ $\bullet$ Every $(4k-2)$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is regular. $\$ $\bullet$ Every $(k^2+2k-1)$-connected line graph $L(G)$ with $\delta(G) \geq k+1$ has $k$ completely independent spanning trees.Comment: 20 pages with 5 figure

Connectivity and spanning trees of graphs

This dissertation focuses on connectivity, edge connectivity and edge-disjoint spanning trees in graphs and hypergraphs from the following aspects.;1. Eigenvalue aspect. Let lambda2(G) and tau( G) denote the second largest eigenvalue and the maximum number of edge-disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of tau(G), Cioaba and Wong conjectured that for any integers d, k ≥ 2 and a d-regular graph G, if lambda 2(G)) \u3c d -- 2k-1d+1 , then tau(G) ≥ k. They proved the conjecture for k = 2, 3, and presented evidence for the cases when k ≥ 4. We propose a more general conjecture that for a graph G with minimum degree delta ≥ 2 k ≥ 4, if lambda2(G) \u3c delta -- 2k-1d+1 then tau(G) ≥ k. We prove the conjecture for k = 2, 3 and provide partial results for k ≥ 4. We also prove that for a graph G with minimum degree delta ≥ k ≥ 2, if lambda2( G) \u3c delta -- 2k-1d +1 , then the edge connectivity is at least k. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on tau(G) and edge connectivity.;2. Network reliability aspect. With graphs considered as natural models for many network design problems, edge connectivity kappa\u27(G) and maximum number of edge-disjoint spanning trees tau(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G. Let kappa\u27(G) = max{lcub}kappa\u27(H) : H is a subgraph of G{rcub}. We present: (i) For each integer k \u3e 0, a characterization for graphs G with the property that kappa\u27(G) ≤ k but for any additional edge e not in G, kappa\u27(G + e) ≥ k + 1. (ii) For any integer n \u3e 0, a characterization for graphs G with |V(G)| = n such that kappa\u27(G) = tau( G) with |E(G)| minimized.;3. Generalized connectivity. For an integer l ≥ 2, the l-connectivity kappal( G) of a graph G is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Let k ≥ 1, a graph G is called (k, l)-connected if kappa l(G) ≥ k. A graph G is called minimally (k, l)-connected if kappal(G) ≥ k but ∀e ∈ E( G), kappal(G -- e) ≤ k -- 1. A structural characterization for minimally (2, l)-connected graphs and some extremal results are obtained. These extend former results by Dirac and Plummer on minimally 2-connected graphs.;4. Degree sequence aspect. An integral sequence d = (d1, d2, ···, dn) is hypergraphic if there is a simple hypergraph H with degree sequence d, and such a hypergraph H is a realization of d. A sequence d is r-uniform hypergraphic if there is a simple r- uniform hypergraph with degree sequence d. It is proved that an r-uniform hypergraphic sequence d = (d1, d2, ···, dn) has a k-edge-connected realization if and only if both di ≥ k for i = 1, 2, ···, n and i=1ndi≥ rn-1r-1 , which generalizes the formal result of Edmonds for graphs and that of Boonyasombat for hypergraphs.;5. Partition connectivity augmentation and preservation. Let k be a positive integer. A hypergraph H is k-partition-connected if for every partition P of V(H), there are at least k(| P| -- 1) hyperedges intersecting at least two classes of P. We determine the minimum number of hyperedges in a hypergraph whose addition makes the resulting hypergraph k-partition-connected. We also characterize the hyperedges of a k-partition-connected hypergraph whose removal will preserve k-partition-connectedness

Mengerin lause ja Tutten nowhere-zero -ongelma

Tässä työssä tutkitaan siirtoverkkoja sekä suuntaamattomien graafien k-virtauksia. Tutustutaan erityisesti Mengerin lauseeseen siirtoverkkojen yhtenäisyydestä sekä Tutten avoimeen ongelmaan, jonka mukaan jokaisella sillattomalla graafilla on nowhere-zero -5-virtaus. Työ alkaa graafien perusmääritelmien esittämisellä. Tämän jälkeen todistetaan siirtoverkoille maksimivirtaus-minimi-irrotus -lause sekä johdetaan samasta todistuksesta vielä Ford–Fulkerson-algoritmi. Lisäksi esitetään Mengerin tulos todistuksineen. Viimeinen luku aloitetaan osoittamalla, että jokaisella suuntaamattomalla graafilla on k-virtaus, jos ja vain jos sillä on Z_k-virtaus. Sen jälkeen tutustutaan Nash-Williamsin lauseeseen, joka yhdistää graafin yhtenäisyyden sekä virittävien puiden lukumäärän. Esitetään vielä Tutten avoin ongelma, joitain samansuuntaisia tuloksia sekä lopuksi todistetaan Seymourin lause, jonka mukaan jokaisella sillattomalla graafilla on nowhere-zero -6-virtaus. Päätulokseen johtavan lemman 4.0.3 olen muokannut ja todistanut itsenäisesti

Cycles and Bases of Graphs and Matroids

The objective of this dissertation is to investigate the properties of cycles and bases in matroids and in graphs. In [62], Tutte defined the circuit graph of a matroid and proved that a matroid is connected if and only if its circuit graph is connected. Motivated by Tutte\u27s result, we introduce the 2nd order circuit graph of a matroid, and prove that for any connected matroid M other than U1,1, the second order circuit graph of M has diameter at most 2 if and only if M does not have a restricted minor isomorphic to U2,6.;Another research conducted in this dissertation is related to the eulerian subgraph problem in graph theory. A graph G is eulerian if G is connected without vertices of odd degrees, and G is supereulerian if G has a spanning eulerian subgraph. In [3], Boesch, Suffey and Tindel raised a problem to determine when a graph is supereulerian, and they remarked that such a problem would be a difficult one. In [55], Pulleyblank confirmed the remark by showing that the problem to determine if a graph is supereulerian, even within planar graphs, is NP-complete. Catlin in [8] introduced a reduction method based on the theory of collapsible graphs to search for spanning eulerian subgraphs in a given graph G. In this dissertation, we introduce the supereulerian width of a graph G, which generalizes the concept of supereulerian graphs, and extends the supereulerian problem to the supereulerian width problem in graphs. Further, we also generalize the concept of collapsible graphs to s-collapsible graphs and develop the reduction method based on the theory of s-collapsible graphs. Our studies extend the collapsible graph theory of Catlin. These are applied to show for any integer n \u3e 2, the complete graph Kn is (n - 3)- collapsible, and so the supereulerian width of Kn is n - 2. We also prove a best possible degree condition for a simple graph to have supereulerian width at least 3.;The number of edge-disjoint spanning trees plays an important role in the design of networks, as it is considered as a measure of the strength of the network. As disjoint spanning trees are disjoint bases in graphic matroids, it is important to study the properties related to the number of disjoint bases in matroids. In this dissertation, we develop a decomposition theory based on the density function of a matroid, and prove a decomposition theorem that partitions the ground set of a matroid M into subsets based on their densities. As applications of the decomposition theorem, we investigate problems related to the properties of disjoint bases in a matroid. We showed that for a given integer k \u3e 0, any matroid M can be embedded into a matroid M\u27 with the same rank (that is, r(M) = r( M\u27)) such that M\u27 has k disjoint bases. Further we determine the minimum value of |E( M\u27)| -- |E(M)| in terms of invariants of M. For a matroid M with at least k disjoint bases, we characterize the set of elements in M such that removing any one of them would still result in a matroid with at least k disjoint bases

A Study of Arc Strong Connectivity of Digraphs

My dissertation research was motivated by Matula and his study of a quantity he called the strength of a graph G, kappa\u27( G) = max{lcub}kappa\u27(H) : H G{rcub}. (Abstract shortened by ProQuest.)