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

Subtrees search, cycle spectra and edge-connectivity structures

In the first part of this thesis, we study subtrees of specified weight in a tree $T$ with vertex weights $c: V(T) \rightarrow \mathbb{N}$. We introduce an overload-discharge method, and discover that there always exists some subtree $S$ whose weight $c(S) := \sum_{v \in V(S)} c(v)$ is close to $\frac{c(T)}{2}$; the smaller the weight $c(T)$ of $T$ is, the smaller difference between $c(S)$ and $\frac{c(T)}{2}$ we can assure. We also show that such a subtree can be found efficiently, namely in linear time. With this tool we prove that every planar Hamiltonian graph $G = (V(G), E(G))$ with minimum degree $\delta \geq 4$ has a cycle of length $k$ for every $k \in \{\lfloor \frac{|V(G)|}{2} \rfloor, \dots, \lceil \frac{|V(G)|}{2} \rceil + 3\}$ with $3 \leq k \leq |V(G)|$. Such a cycle can be found in linear time if a Hamilton cycle of the graph is given. In the second part of the thesis, we present three cut trees of a graph, each of them giving insights into the edge-connectivity structure. All three cut trees have in common that they cover a given binary symmetric irreflexive relation on the vertex set of the graph, while generalizing Gomory-Hu trees. With these cut trees we show the following: (i) every simple graph $G$ with $\delta \geq 5$ or with edge-connectivity $\lambda \geq 4$ or with vertex-connectivity $\kappa \geq 3$ contains at least $\frac{1}{24}\delta |V(G)|$ pendant pairs, where a pair of vertices $\{v, w\}$ is pendant if $\lambda_G(v,w) = \min\{d_G(v), d_G(w)\}$; (ii) every simple graph $G$ satisfying $\delta > 0$ has $O(|V(G)|/\delta)$ $\delta$-edge-connected components, and there are only $O(|V(G)|)$ edges left if these components are contracted; (iii) given a simple graph $G$ satisfying $\delta > 0$, one can find some vertex subsets in near-linear time such that all non-trivial min-cuts are preserved, and $O(|V(G)|/\delta)$ vertices and $O(|V(G)|)$ edges remain when these vertex subsets are contracted.Im ersten Teil dieser Dissertation untersuchen wir Teilbäume eines Baumes $T$ mit vorgegebenen Knotengewichten $c: V(T) \rightarrow \mathbb{N}$. Wir führen eine Overload-Discharge-Methode ein, und zeigen, dass es immer einen Teilbaum $S$ gibt, dessen Gewicht $c(S) := \sum_ {v \in V (S)} c(v)$ nahe $\frac{c(T)}{2}$ liegt. Je kleiner das Gewicht $c(T)$ von $T$ ist, desto geringer ist dabei die Differenz zwischen $c(S)$ und $\frac{c(T)}{2}$, die wir sicherstellen können. Wir zeigen auch, dass ein solcher Teilbaum effizient, nämlich in Linearzeit, berechnet werden kann. Unter Ausnutzung dieser Methode beweisen wir, dass jeder planare hamiltonsche Graph $G = (V(G), E(G))$ mit Mindestgrad $\delta \geq 4$ einen Kreis der Länge $k$ für jedes $k \in \{\lfloor \frac{|V(G)|}{2} \rfloor, \dots, \lceil \frac{|V(G)|}{2} \rceil + 3\}$ mit $3 \leq k \leq |V (G)|$ enthält. Dieser kann in Linearzeit berechnet werden, falls ein Hamilton-Kreis des Graphen bekannt ist. Im zweiten Teil der Dissertation stellen wir drei Schnittbäume eines Graphen vor, von denen jeder Einblick in die Kantenzusammenhangsstruktur des Graphen gibt. Allen drei Schnittbäumen ist gemeinsam, dass sie eine bestimmte binäre symmetrische irreflexive Relation auf der Knotenmenge des Graphen überdecken; die Bäume können als Verallgemeinerungen von Gomory-Hu-Bäumen aufgefasst werden. Die Schnittbäume implizieren folgende Aussagen: (i) Jeder schlichte Graph $G$, der $\delta \geq 5$ oder Kantenzusammenhang $\lambda \geq 4$ oder Knotenzusammenhang $\kappa \geq 3$ erfüllt, enthält mindestens $\frac{1}{24} \delta |V(G)|$ zusammengehörige Paare, wobei ein Paar von Knoten $\{v, w \}$ zusammengehörig ist, falls $\lambda_G (v, w) = \min \{d_G(v), d_G(w)\}$ ist. (ii) Jeder schlichte Graph $G$ mit $\delta > 0$ hat $O(|V (G)| / \delta)$ $\delta$-kantenzusammenhängende Komponenten, und es verbleiben lediglich $O(|V (G)|)$ Kanten, wenn diese Komponenten kontrahiert werden. (iii) Für jeden schlichten Graphen $G$ mit $\delta > 0$ sind Knotenmengen derart effizient berechenbar, dass alle nicht trivialen minimalen Schnitte erhalten bleiben, und $O(|V(G)| / \delta)$ Knoten und $O(|V(G)|)$ Kanten verbleiben, wenn diese Knotenmengen kontrahiert werden

Tomescu\u27s Graph Coloring Conjecture for -Connected Graphs

Let PG(k) be the number of proper k-colorings of a finite simple graph G. Tomescu\u27s conjecture, which was recently solved by Fox, He, and Manners, states that PG(k)k!(k-1)(n – k) for all connected graphs G on n vertices with chromatic number k≥4. In this paper, we study the same problem with the additional constraint that G is ℓ-connected. For 2-connected graphs G, we prove a tight bound PG(k)≤(k – 1)!((k – 1)(n – k+1) + ( - 1)n – k) and show that equality is only achieved if G is a k-clique with an ear attached. For ℓ≥3, we prove an asymptotically tight upper bound PG(k)≤k!(k-1)n-l-k+1+O((k – 2)n ) and provide a matching lower bound construction. For the ranges k≥ℓ or ℓ ≥ (k-2)(k-1)+ 1 we further find the unique graph maximizing . We also consider generalizing ℓ-connected graphs to connected graphs with minimum degree δ