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

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}$G$} is Hamilton-connected if for any pair of distinct vertices {\color{black}$u, v \in V(G)$}, {\color{black}$G$} has a spanning $(u,v)$-path; {\color{black}$G$} is 1-hamiltonian if for any vertex subset $S \subseteq {\color{black}V(G)}$ with $|S| \le 1$, $G - S$ has a spanning cycle. Let $\delta(G)$, $\alpha\u27(G)$ and $L(G)$ denote the minimum degree, the matching number and the line graph of a graph $G$, respectively. The following result is obtained. {\color{black} Let $G$ be a simple graph} with $|E(G)| \ge 3$. If $\delta(G) \geq \alpha\u27(G)$, then each of the following holds. \\ (i) $L(G)$ is Hamilton-connected if and only if $\kappa(L(G))\ge 3$. \\ (ii) $L(G)$ is 1-hamiltonian if and only if $\kappa(L(G))\ge 3$. %==========sp For a graph $G$, an integer $s \ge 0$ and distinct vertices $u, v \in V(G)$, an $(s; u, v)$-path-system of $G$ is a subgraph $H$ consisting of $s$ internally disjoint $(u,v)$-paths. The spanning connectivity $\kappa^*(G)$ is the largest integer $s$ such that for any $k$ with $0 \le k \le s$ and for any $u, v \in V(G)$ with $u \neq v$, $G$ has a spanning $(k; u,v)$-path-system. It is known that $\kappa^*(G) \le \kappa(G)$, and determining if $\kappa^*(G) \u3e 0$ is an NP-complete problem. A graph $G$ is maximally spanning connected if $\kappa^*(G) = \kappa(G)$. Let $msc(G)$ and $s_k(G)$ be the smallest integers $m$ and $m\u27$ such that $L^m(G)$ is maximally spanning connected and $\kappa^*(L^{m\u27}(G)) \ge k$, 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 $msc(G)$ and $s_k(G)$, and characterized the extremal graphs reaching the upper bounds. %==============st For integers $s \ge 0$ and $t \ge 0$, a graph $G$ is $(s,t)$-supereulerian if for any disjoint edge sets $X, Y \subseteq E(G)$ with $|X|\le s$ and $|Y|\le t$, $G$ has a spanning closed trail that contains $X$ and avoids $Y$. Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is $(0,0)$-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 $G$ on $n$ vertices with $\delta(G) \ge \frac{n}{5} -1$, when $n$ is sufficiently large, is $(0,0)$-supereulerian or is contractible to $K_{2,3}$. We prove the following for any nonnegative integers $s$ and $t$. \\ (i) For any real numbers $a$ and $b$ with $0 \u3c a \u3c 1$, there exists a family of finitely many graphs \F(a,b;s,t) such that if $G$ is a simple graph on $n$ vertices with $\kappa\u27(G) \ge t+2$ and $\delta(G) \ge an + b$, then either $G$ is $(s,t)$-supereulerian, or $G$ is contractible to a member in \F(a,b;s,t). \\ (ii) Let $\ell K_2$ denote the connected loopless graph with two vertices and $\ell$ parallel edges. If $G$ is a simple graph on $n$ vertices with $\kappa\u27(G) \ge t+2$ and $\delta(G) \ge \frac{n}{2}-1$, then when $n$ is sufficiently large, either $G$ is $(s,t)$-supereulerian, or for some integer $j$ with $t+2 \le j \le s+t$, $G$ is contractible to a $j K_2$. %==================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\}: $\kappa\u27(G) \ge a$ and $\delta(G) \ge b\}$, and proposed a few problems to determine \cp(a,b) with $b \ge a \ge 4$ 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 $ess\u27(G)$ denote the essential edge-connectivity of a graph $G$, and define \cp\u27(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: $ess\u27(G) \ge a$ and $\delta(G) \ge b\}$. 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 $b \ge 1$, \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

Cycles, Disjoint Spanning Trees and Orientations of Graphs

A graph G is hamiltonian-connected if any two of its vertices are connected by a Hamilton path (a path including every vertex of G); and G is s-hamiltonian-connected if the deletion of any vertex subset with at most s vertices results in a hamiltonian-connected graph. We prove that the line graph of a (t + 4)-edge-connected graph is (t + 2)-hamiltonian-connected if and only if it is (t + 5)-connected, and for s ≥ 2 every (s + 5)-connected line graph is s-hamiltonian-connected.;For integers l and k with l \u3e 0, and k ≥ 0, Ch( l, k) denotes the collection of h-edge-connected simple graphs G on n vertices such that for every edge-cut X with 2 ≤ |X| ≤ 3, each component of G -- X has at least (n -- k)/l vertices. We prove that for any integer k \u3e 0, there exists an integer N = N( k) such that for any n ≥ N, any graph G ∈ C2(6, k) on n vertices is supereulerian if and only if G cannot be contracted to a member in a well characterized family of graphs.;An orientation of an undirected graph G is a mod (2 p + 1)-orientation if under this orientation, the net out-degree at every vertex is congruence to zero mod 2p + 1. A graph H is mod (2p + 1)-contractible if for any graph G that contains H as a subgraph, the contraction G/H has a mod (2p + 1)-orientation if and only if G has a mod (2p + 1)-orientation (thus every mod (2p + 1)-contractible graph has a mod (2p + 1)-orientation). Jaeger in 1984 conjectured that every (4p)-edge-connected graph has a mod (2p + 1)-orientation. It has also been conjectured that every (4p + 1)-edge-connected graph is mod (2 p + 1)-contractible. We investigate graphs that are mod (2 p + 1)-contractible, and as applications, we prove that a complete graph Km is (2p + 1)-contractible if and only if m ≥ 4p + 1; that every (4p -- 1)-edge-connected K4-minor free graph is mod (2p + 1)-contractible, which is best possible in the sense that there are infinitely many (4p -- 2)-edge-connected K4-minor free graphs that are not mod (2p + 1)-contractible; and that every (4p)-connected chordal graph is mod (2p + 1)-contractible. We also prove that the above conjectures on line graphs would imply the truth of the conjectures in general, and that if G has a mod (2p + 1)-orientation and delta(G) ≥ 4p, then L(G) also has a mod (2p + 1)-orientation.;The design of an n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties. A nonincreasing sequence d = ( d1, d2, ···, dn) is graphic if there is a simple graph G with degree sequence d. It is proved that for a positive integer k, a graphic nonincreasing sequence d has a simple realization G which has k-edge-disjoint spanning trees if and only if either both n = 1 and d1 = 0, or n ≥ 2 and both dn ≥ k and i=1n di ≥ 2k(n -- 1).;We investigate the emergence of specialized groups in a swarm of robots, using a simplified version of the stick-pulling problem [56], where the basic task requires the collaboration of two robots in asymmetric roles. We expand our analytical model [57] and identify conditions for optimal performance for a swarm with any number of species. We then implement a distributed adaptation algorithm based on autonomous performance evaluation and parameter adjustment of individual agents. While this algorithm reliably reaches optimal performance, it leads to unbounded parameter distributions. Results are improved by the introduction of a direct parameter exchange mechanism between selected high- and low-performing agents. The emerging parameter distributions are bounded and fluctuate between tight unimodal and bimodal profiles. Both the unbounded optimal and the bounded bimodal distributions represent partitions of the swarm into two specialized groups

On Spanning Disjoint Paths in Line Graphs

Spanning connectivity of graphs has been intensively investigated in the study of interconnection networks (Hsu and Lin, Graph Theory and Interconnection Networks, 2009). For a graph G and an integer s \u3e 0 and for u,v∈V(G)u,v∈V(G) with u ≠ v, an (s; u, v)-path-system of G is a subgraph H consisting of s internally disjoint (u,v)-paths. A graph G is spanning s-connected if for any u,v∈V(G)u,v∈V(G) with u ≠ v, G has a spanning (s; u, v)-path-system. The spanning connectivity κ*(G) of a graph G is the largest integer s such that G has a spanning (k; u, v)-path-system, for any integer k with 1 ≤ k ≤ s, and for any u,v∈V(G)u,v∈V(G) with u ≠ v. An edge counter-part of κ*(G), defined as the supereulerian width of a graph G, has been investigated in Chen et al. (Supereulerian graphs with width s and s-collapsible graphs, 2012). In Catlin and Lai (Graph Theory, Combinatorics, and Applications, vol. 1, pp. 207–222, 1991) proved that if a graph G has 2 edge-disjoint spanning trees, and if L(G) is the line graph of G, then κ*(L(G)) ≥ 2 if and only if κ(L(G)) ≥ 3. In this paper, we extend this result and prove that for any integer k ≥ 2, if G 0, the core of G, has k edge-disjoint spanning trees, then κ*(L(G)) ≥ k if and only if κ(L(G)) ≥ max{3, k}