Collapsible graphs and reductions of line graphs

A graph G is collapsible if for every even subset X ⊆ V ( G ) , G has a subgraph such that G − E ( Γ ) is connected and the set of odd-degree vertices of Γ is X . A graph obtained by contracting all the non-trivial collapsible subgraphs of G is called the reduction of G . In this paper, we characterize graphs of diameter two in terms of collapsible subgraphs and investigate the relationship between the line graph of the reduction and the reduction of the line graph. Our results extend former results in [H.-J. Lai, Reduced graph of diameter two, J. Graph Theory 14 (1) (1990) 77–87], and in [P.A. Catlin, Iqblunnisa, T.N. Janakiraman, N. Srinivasan, Hamilton cycles and closed trails in iterated line graphs, J. Graph Theory 14 (1990) 347–364]. ]

On dominating and spanning circuits in graphs

An eulerian subgraph of a graph is called a circuit. As shown by Harary and Nash-Williams, the existence of a Hamilton cycle in the line graph L(G) of a graph G is equivalent to the existence of a dominating circuit in G, i.e., a circuit such that every edge of G is incident with a vertex of the circuit. Important progress in the study of the existence of spanning and dominating circuits was made by Catlin, who defined the reduction of a graph G and showed that G has a spanning circuit if and only if the reduction of G has a spanning circuit. We refine Catlin's reduction technique to obtain a result which contains several known and new sufficient conditions for a graph to have a spanning or dominating circuit in terms of degree-sums of adjacent vertices. In particular, the result implies the truth of the following conjecture of Benhocine et al.: If G is a connected simple graph of order n such that every cut edge of G is incident with a vertex of degree 1 and d(u)+d(v) > 2(1/5n-1) for every edge uv of G, then, for n sufficiently large, L(G) is hamiltonian

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

For a graph H , let σ t ( H ) = min { Σ i = 1 t d H ( v i ) | { v 1 , v 2 , … , v t } is an independent set in H } and let U t ( H ) = min { | ⋃ i = 1 t N H ( v i ) | | { v 1 , v 2 , ⋯ , v t } is an independent set in H } . We show that for a given number ϵ and given integers p ≥ t \u3e 0 , k ∈ { 2 , 3 } and N = N ( p , ϵ ) , if H is a k -connected claw-free graph of order n \u3e N with δ ( H ) ≥ 3 and its Ryjác̆ek’s closure c l ( H ) = L ( G ) , and if d t ( H ) ≥ t ( n + ϵ ) ∕ p where d t ( H ) ∈ { σ t ( H ) , U t ( H ) } , then either H is Hamiltonian or G , the preimage of L ( G ) , can be contracted to a k -edge-connected K 3 -free graph of order at most max { 4 p − 5 , 2 p + 1 } and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large: (i) If k = 2 , δ ( H ) ≥ 3 , and for a given t ( 1 ≤ t ≤ 4 ), then either H is Hamiltonian or c l ( H ) = L ( G ) where G is a graph obtained from K 2 , 3 by replacing each of the degree 2 vertices by a K 1 , s ( s ≥ 1 ). When t = 4 and d t ( H ) = σ 4 ( H ) , this proves a conjecture in Frydrych (2001). (ii) If k = 3 , δ ( H ) ≥ 24 , and for a given t ( 1 ≤ t ≤ 10 ) d t ( H ) \u3e t ( n + 5 ) 10 , then H is Hamiltonian. These bounds on d t ( H ) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved σ t and U t for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max { 4 p − 5 , 2 p + 1 } are fixed for given p , improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer

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

Claw -free graphs and line graphs

The research of my dissertation is motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Tutte that every 4-edge-connected graph has a no-where-zero 3-flow. Towards the hamiltonian line graph problem, we proved that every 3-connected N2-locally connected claw-free graph is hamiltonian, which was conjectured by Ryjacek in 1990; that every 4-connected line graph of an almost claw free graph is hamiltonian connected, and that every triangularly connected claw-free graph G with |E( G)| ≥ 3 is vertex pancyclic. Towards the second conjecture, we proved that every line graph of a 4-edge-connected graph is Z 3-connected

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

Supereulerian Properties in Graphs and Hamiltonian Properties in Line Graphs

Following the trend initiated by Chvatal and Erdos, using the relation of independence number and connectivity as sufficient conditions for hamiltonicity of graphs, we characterize supereulerian graphs with small matching number, which implies a characterization of hamiltonian claw-free graph with small independence number.;We also investigate strongly spanning trailable graphs and their applications to hamiltonian connected line graphs characterizations for small strongly spanning trailable graphs and strongly spanning trailable graphs with short longest cycles are obtained. In particular, we have found a graph family F of reduced nonsupereulerian graphs such that for any graph G with kappa\u27(G) ≥ 2 and alpha\u27( G) ≤ 3, G is supereulerian if and only if the reduction of G is not in F..;We proved that any connected graph G with at most 12 vertices, at most one vertex of degree 2 and without vertices of degree 1 is either supereulerian or its reduction is one of six exceptional cases. This is applied to show that if a 3-edge-connected graph has the property that every pair of edges is joined by a longest path of length at most 8, then G is strongly spanning trailable if and only if G is not the wagner graph.;Using charge and discharge method, we prove that every 3-connected, essentially 10-connected line graph is hamiltonian connected. We also provide a unified treatment with short proofs for several former results by Fujisawa and Ota in [20], by Kaiser et al in [24], and by Pfender in [40]. New sufficient conditions for hamiltonian claw-free graphs are also obtained

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 $\kappa\u27(G)$ and $\delta(G)$ be the edge-connectivity and the minimum degree of a graph $G$, respectively. 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 $Y$. This dissertation is devoted to providing some results on $(s,t)$-supereulerian graphs and supereulerian hypergraphs. In Chapter 2, we determine the value of the smallest integer $j(s,t)$ such that every $j(s,t)$-edge-connected graph is $(s,t)$-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 $(s,t)$-supereulerian graphs when $t \ge 3$ in terms of edge-connectivities, and show that when $t \ge 3$, $(s,t)$-supereulerianicity is polynomially determinable. In Chapter 3, for a subset $Y \subseteq E(G)$ with $|Y|\le \kappa\u27(G)-1$, a necessary and sufficient condition for $G-Y$ to be a contractible configuration for supereulerianicity is obtained. We also characterize the $(s,t)$-supereulerianicity of $G$ when $s+t\le \kappa\u27(G)$. These results are applied to show that if $G$ is $(s,t)$-supereulerian with $\kappa\u27(G)=\delta(G)\ge 3$, then for any permutation $\alpha$ on the vertex set $V(G)$, the permutation graph $\alpha(G)$ is $(s,t)$-supereulerian if and only if $s+t\le \kappa\u27(G)$. For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Let $i_{s,t}(G)$ and $h_s(G)$ denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is $(s,t)$-supereulerian and $s$-Hamiltonian, respectively. In Chapter 4, for a simple graph $G$, we establish upper bounds for $i_{s,t}(G)$ and $h_s(G)$. Specifically, the upper bound for the $s$-Hamiltonian index $h_s(G)$ 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 $G$ is Hamiltonian if and only if $G$ 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 $H$ defined by Rodr\\u27iguez in 2002, let $\lambda_2(H)$ be the second largest adjacency eigenvalue of $H$. In Chapter 6, we prove that for an integer $k$ and a $r$-uniform hypergraph $H$ of order $n$ with $r\ge 4$ even, the minimum degree $\delta\ge k\ge 2$ and $k\neq r+2$, if $\lambda_2(H)\le (r-1)\delta-\frac{r^2(k-1)n}{4(r+1)(n-r-1)}$, then $H$ is $k$-edge-connected. %$\kappa\u27(H)\ge k$. 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 $(s,t)$-supereulerianicity of hypergraphs is another interesting topic to be investigated in the future

Degree Conditions for Hamiltonian Properties of Claw-free Graphs

This thesis contains many new contributions to the field of hamiltonian graph theory, a very active subfield of graph theory. In particular, we have obtained new sufficient minimum degree and degree sum conditions to guarantee that the graphs satisfying these conditions, or their line graphs, admit a Hamilton cycle (or a Hamilton path), unless they have a small order or they belong to well-defined classes of exceptional graphs. Here, a Hamilton cycle corresponds to traversing the vertices and edges of the graph in such a way that all their vertices are visited exactly once, and we return to our starting vertex (similarly, a Hamilton path reflects a similar way of traversing the graph, but without the last restriction, so we might terminate at a different vertex). In Chapter 1, we presented an introduction to the topics of this thesis together with Ryjáček’s closure for claw-free graphs, Catlin’s reduction method, and the reduction of the core of a graph. In Chapter 2, we found the best possible bounds for the minimum degree condition and the minimum degree sums condition of adjacent vertices for traceability of 2-connected claw-free graph, respectively. In addition, we decreased these lower bounds with one family of well characterized exceptional graphs. In Chapter 3, we extended recent results about the conjecture of Benhocine et al. and results about the conjecture of Z.-H Chen and H.-J Lai. In Chapters 4, 5 and 6, we have successfully tried to unify and extend several existing results involving the degree and neighborhood conditions for the hamiltonicity and traceability of 2-connected claw-free graphs. Throughout this thesis, we have investigated the existence of Hamilton cycles and Hamilton paths under different types of degree and neighborhood conditions, including minimum degree conditions, minimum degree sum conditions on adjacent pairs of vertices, minimum degree sum conditions over all independent sets of t vertices of a graph, minimum cardinality conditions on the neighborhood union over all independent sets of t vertices of a graph, as well minimum cardinality conditions on the neighborhood union over all t vertex sets of a graph. Despite our new contributions, many problems and conjectures remain unsolved