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

Spanning Trails and Spanning Trees

There are two major parts in my dissertation. One is based on spanning trail, the other one is comparing spanning tree packing and covering.;The results of the spanning trail in my dissertation are motivated by Thomassen\u27s Conjecture that every 4-connected line graph is hamiltonian. Harary and Nash-Williams showed that the line graph L( G) is hamiltonian if and only if the graph G has a dominating eulerian subgraph. Also, motivated by the Chinese Postman Problem, Boesch et al. introduced supereulerian graphs which contain spanning closed trails. In the spanning trail part of my dissertation, I proved some results based on supereulerian graphs and, a more general case, spanning trails.;Let alpha(G), alpha\u27(G), kappa( G) and kappa\u27(G) denote the independence number, the matching number, connectivity and edge connectivity of a graph G, respectively. First, we discuss the 3-edge-connected graphs with bounded edge-cuts of size 3, and prove that any 3-edge-connected graph with at most 11 edge cuts of size 3 is supereulerian, which improves Catlin\u27s result. Second, having the idea from Chvatal-Erdos Theorem which states that every graph G with kappa(G) ≥ alpha( G) is hamiltonian, we find families of finite graphs F 1 and F2 such that if a connected graph G satisfies kappa\u27(G) ≥ alpha(G) -- 1 (resp. kappa\u27(G) ≥ 3 and alpha\u27( G) ≤ 7), then G has a spanning closed trail if and only if G is not contractible to a member of F1 (resp. F2). Third, by solving a conjecture posed in [Discrete Math. 306 (2006) 87-98], we prove if G is essentially 4-edge-connected, then for any edge subset X0 ⊆ E(G) with |X0| ≤ 3 and any distinct edges e, e\u27 2 ∈ E(G), G has a spanning ( e, e\u27)-trail containing all edges in X0.;The results on spanning trees in my dissertation concern spanning tree packing and covering. We find a characterization of spanning tree packing and covering based on degree sequence. Let tau(G) be the maximum number of edge-disjoint spanning trees in G, a(G) be the minimum number of spanning trees whose union covers E(G). We prove that, given a graphic sequence d = (d1, d2···dn) (d1 ≥ d2 ≥···≥ dn) and integers k2 ≥ k1 \u3e 0, there exists a simple graph G with degree sequence d satisfying k 1 ≤ tau(G) ≤ a(G) ≤ k2 if and only if dn ≥ k1 and 2k1(n -- 1) ≤ Sigmani =1 di ≤ 2k2( n -- 1 |I| -- 1) + 2Sigma i∈I di, where I = {lcub}i : di \u3c k2{rcub}

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

Group Colorability and Hamiltonian Properties of Graphs

The research of my dissertation was motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Matthews and Sumner that every 4-connected claw-free graph is hamiltonian. Towards the hamiltonian line graph problem, we proved that every 3-edge-connected, essentially 4-edge-connected graph G has a spanning eulerian subgraph, if for every pair of adjacent vertices u and v, dG(u) + dG(v) ≥ 9. A straight forward corollary is that every 4-connected, essentially 6-connected line graph with minimum degree at least 7 is hamiltonian.;We also investigate graphs G such that the line graph L(G) is hamiltonian connected when L( G) is 4-connected. Ryjacek and Vrana recently further conjectured that every 4-connected line graph is hamiltonian-connected. In 2001, Kriesell proved that every 4-connected line graph of a claw free graph is hamiltonian connected. Recently, Lai et al showed that every 4-connected line graph of a quasi claw free graph is hamiltonian connected, and that every 4-connected line graph of an almost claw free graph is hamiltonian connected. In 2009, Broersma and Vumer discovered the P3-dominating (P3D) graphs as a superfamily that properly contains all quasi claw free graphs, and in particular, all claw-free graphs. Here we prove that every 4-connected line graph of a P3D graph is hamiltonian connected, which extends several former results in this area.;R. Gould [15] asked what natural graph properties of G and H are sufficient to imply that the product of G and H is hamiltonian. We first investigate the sufficient and necessary conditions for G x H being hamiltonian or traceable when G is a hamiltonian graph and H is a tree. Then we further investigate sufficient and necessary conditions for G x H being hamiltonian connected, or edge-pancyclic, or pan-connected.;The problem of group colorings of graphs is also investigated in this dissertation. Group coloring was first introduced by Jeager et al. [21]. They introduced a concept of group connectivity as a generalization of nowhere-zero flows. They also introduced group coloring as a dual concept to group connectivity. Prior research on group chromatic number was restricted to simple graphs, and considered only Abelian groups in the definition of chi g(G). The behavior of group coloring for multigraphs is different to that of simple graphs. Thus we extend the definition of group coloring by considering general groups (both Abelian groups and non-Abelian groups), and investigate the properties of chig for multigraphs by proving an analogue to Brooks\u27 Theorem

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

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306–315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K5 − E(C4), where C4 is an cycle of length 4 in K5) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G)) ≥ 3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306–315] that every 4-connected hourglass free line graph is hamiltonian connected