Eulerian Subgraph Research Articles

Continuous fiber printing path generation based on Euler diagram division

Continuous fiber printed structures are widely used in aerospace applications because of their excellent performance and high design freedom. Existing path planning methods optimize the continuous fiber printing direction by stress lines or discrete angles, but these methods are far from perfect printing paths and lack planning for connectivity relations. In this study, a planning method for continuous fiber printing paths based on constructing Eulerian circuits from a weighted undirected graph is proposed. The principal stress lines are transformed into a weighted undirected graph, which divides the Eulerian subgraph by splitting the common nodes to plan the continuous fiber printing path circuit. Subsequently, the printing structure parameters are optimized based on simulation analysis. The experimental results, taking MBB (Messerschmitt-Bolkow-Blohm) beams as an example, show that the printing path designed in this paper improves the structural strength by 52.9% and the structural stiffness by 81.7% compared with the conventional printing path.

Cycle algebras and polytopes of matroids

Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes.

Constructing featured supereulerian graph

A graph is supereulerian if it has a spanning eulerian subgraph. In this paper we construct a supereulerian graph controlling its matching number, minimum degree and edge connectivity, where δ(G) < α′(G)

On computing the Hamiltonian index of graphs

For an integer r≥0 the r-th iterated line graphLr(G) of a graph G is defined by: (i) L0(G)=G and (ii) Lr(G)=L(L(r−1)(G)) for r>0, where L(G) denotes the line graph of G. The Hamiltonian Indexh(G) of G is the smallest r such that Lr(G) has a Hamiltonian cycle [Chartrand, 1968]. Checking if h(G)=k is NP-hard for any fixed integer k≥0 even for subcubic graphs G [Ryjáček et al., 2011]. We study the parameterized complexity of this problem with the parameter treewidth, tw(G), and show that we can find h(G) in time1O⋆((1+2(ω+3))tw(G)) where ω is the matrix multiplication exponent. Prior work on computing h(G) includes various O⋆(2O(tw(G)))-time algorithms for checking if h(G)=0 holds; i.e., whether G has a Hamiltonian Cycle [Cygan et al., FOCS 2011; Bodlaender et al., Inform. Comput., 2015; Fomin et al., JACM 2016]; an O⋆(tw(G)O(tw(G)))-time algorithm for checking if h(G)=1 holds; i.e., whether L(G) has a Hamiltonian Cycle [Lampis et al., Discrete Appl. Math., 2017]; and, most recently, an O⋆((1+2(ω+3))tw(G))-time algorithm for checking if h(G)=1 holds [Misra et al., CSR 2019]. Our algorithm for computing h(G) generalizes these results.The NP-hard Eulerian Steiner Subgraph problem takes as input a graph G and a specified subset K of terminal vertices of G and asks if G has an Eulerian2 subgraph H containing all the terminals. A key ingredient of our algorithm for finding h(G) is an algorithm which solves Eulerian Steiner Subgraph in O⋆((1+2(ω+3))tw(G)) time. To the best of our knowledge this is the first FPT algorithm for Eulerian Steiner Subgraph. Prior work on the special case of finding a spanning Eulerian subgraph (i.e., with K=V(G)) includes a polynomial-time algorithm for series-parallel graphs [Richey et al., 1985] and an O⋆(2O(n))-time algorithm for planar graphs on n vertices [Sau and Thilikos, 2010]. Our algorithm for Eulerian Steiner Subgraph generalizes both these results.

Eulerian ideals

Let G be a simple graph and , where is the homomorphism that sends an edge to the product of its vertices. The ideal is Cohen–Macaulay, one-dimensional and binomial. If G is bipartite, it is known that the Castelnuovo–Mumford regularity of is equal to the maximum cardinality of a set of edges having no more than half of the edges of any Eulerian subgraph of G. Here, with respect to the grevlex order associated to an ordering of the edge set of G, we describe a Gröbner basis for , and we characterize the standard monomials of the ideal in terms of even sets of vertices marked with a parity. Using these results, we give a combinatorial interpretation of the degree of , via the set of even sets of vertices of G; and we show that the Castelnuovo–Mumford regularity of , for any graph, is the maximum cardinality of a set of edges having no more than half of the edges of any even Eulerian subgraph of G or, equivalently, the maximum cardinality of a minimum fixed parity T-join.

On hamiltonian line graphs of hypergraphs

AbstractA graph is supereulerian if it has a spanning eulerian subgraph. Harary and Nash‐Williams in 1968 proved that the line graph of a graph is hamiltonian if and only if has a dominating eulerian subgraph, 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. Utilizing the notion of partition‐connectedness of hypergraphs introduced by Frank, Király, 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.

Forbidden subgraphs for supereulerian and hamiltonian graphs

A graph is called supereulerian if it has a spanning eulerian subgraph. A graph is said to be hamiltonian if it has a spanning cycle. A nontrivial path is called a branch if it has only internal vertices of degree two and end vertices of degree not two. Let S be a set of branches of G, then S is called a branch cut if G−S has more components than G. A minimal branch cut is called a branch-bond. In this paper, we characterize one or pairs of those forbidden subgraphs that force a 2-edge-connected graph satisfying that every odd branch-bond has an edge branch to be supereulerian. We also characterize one or pairs of those forbidden subgraphs that force a 2-connected supereulerian graph to be hamiltonian.

Supereulerian Graphs with Constraints on the Matching Number and Minimum Degree

A graph is supereulerian if it has a spanning eulerian subgraph. We show that a connected simple graph G with $$n = |V(G)| \ge 2$$ and $$\delta (G) \ge \alpha '(G)$$ is supereulerian if and only if $$G \ne K_{1,n-1}$$ if n is even or $$G \ne K_{2, n-2}$$ if n is odd. Consequently, every connected simple graph G with $$\delta (G) \ge \alpha '(G)$$ has a hamiltonian line graph.

A refinement of choosability of graphs

Assume k is a positive integer, λ={k1,k2,…,kq} is a partition of k and G is a graph. A λ-assignment of G is a k-assignment L of G such that the colour set ⋃v∈V(G)L(v) can be partitioned into q subsets C1∪C2…∪Cq and for each vertex v of G, |L(v)∩Ci|=ki. We say G is λ-choosable if for each λ-assignment L of G, G is L-colourable. It follows from the definition that if λ={k}, then λ-choosable is the same as k-choosable, if λ={1,1,…,1}, then λ-choosable is equivalent to k-colourable. For the other partitions of k sandwiched between {k} and {1,1,…,1} in terms of refinements, λ-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions λ,λ′ of k, every λ-choosable graph is λ′-choosable if and only if λ′ is a refinement of λ. Then we study λ-choosability of special families of graphs. The Four Colour Theorem says that every planar graph is {1,1,1,1}-choosable. A very recent result of Kemnitz and Voigt implies that for any partition λ of 4 other than {1,1,1,1}, there is a planar graph which is not λ-choosable. We observe that, in contrast to the fact that there are non-4-choosable 3-chromatic planar graphs, every 3-chromatic planar graph is {1,3}-choosable, and that if G is a planar graph whose dual G⁎ has a connected spanning Eulerian subgraph, then G is {2,2}-choosable. We prove that if n is a positive even integer, λ is a partition of n−1 in which each part is at most 3, then Kn is edge λ-choosable. Finally we study relations between λ-choosability of graphs and colouring of signed graphs and generalized signed graphs. A conjecture of Máčajová, Raspaud and Škoviera that every planar graph is signed 4-colcourable is recently disproved by Kardoš and Narboni. We prove that every signed 4-colourable graph is weakly 4-choosable, and every signed Z4-colourable graph is {1,1,2}-choosable. The later result combined with the above result of Kemnitz and Voigt disproves a conjecture of Kang and Steffen that every planar graph is signed Z4-colourable. We shall show that a graph constructed by Wegner in 1973 is also a counterexample to Kang and Steffen's conjecture, and present a new construction of a non-{1,3}-choosable planar graphs.

Spanning Eulerian Subgraphs of Large Size

A graph $$G=(V(G), E(G))$$ is supereulerian if it has a spanning Eulerian subgraph. Let $$\ell (G)$$ be the maximum number of edges of spanning Eulerian subgraphs of a graph G. Motivated by a conjecture due to Catlin on supereulerian graphs, it was shown that if G is an r-regular supereulerian graph, then $$\ell (G)\ge \frac{2}{3}|E(G)|$$ when $$r\ne 5$$ , and $$\ell (G)> \frac{3}{5}|E(G)|$$ when $$r=5$$ . In this paper we improve the coefficient and prove that if G is a 5-regular supereulerian graph, then $$\ell (G)\ge \frac{19}{30}|E(G)|+\frac{4}{3}$$ . For this, we first show that each graph G with maximum degree at most 3 has a matching with at least $$\frac{2}{7}|E(G)|$$ edges and this bound is sharp. Moreover, we show that Catlin’s conjecture holds for claw-free graphs having no vertex of degree 4. Indeed, Catlin’s conjecture does not hold for claw-free graphs in general.

A 4/3-approximation for TSP on cubic 3-edge-connected graphs

For a 3-edge-connected cubic graph G=(V,E), we give an algorithm to construct a connected Eulerian subgraph of 2G using at most ⌊4|V|∕3⌋ edges.

Integer flows and subgraph covers

Integer flows and subgraph covers are two important subjects in graph theory. They are closely related to the famous four color problem, which is equivalent to the integer 4-flow problem in planar graphs. A graph has an integer $k$-flow if for some orientation, there is a function from its edge set to an abelian group such that at each vertex the sum of the values on the edges into the vertex is equal to the sum of the values on the edges out of the vertex. Integer flows are related to several well-known problems in other fields of mathematics, such as lonely runner problem in combinatorics, diophantine approximation in number theory, view obstruction in geometry and additive basis in linear vector space. Four color problem is also equivalent to the problem of Eulerian subgraph covering of planar graphs: Every 2-edge-connected planar graph has three Eulerian subgraphs covering each edge precisely twice. The well-known Fulkerson conjecture claims that every 2-edge-connected graph (not necessary to be planar) has six Eulerian subgraphs covering each edge precisely four times. In this paper, we shall give a brief survey of the two subjects and other related problems.

Exploring Hierarchies in Online Social Networks

Social hierarchy (i.e., pyramid structure of societies) is a fundamental concept in sociology and social network analysis. The importance of social hierarchy in a social network is that the topological structure of the social hierarchy is essential in both shaping the nature of social interactions between individuals and unfolding the structure of the social networks. The social hierarchy found in a social network can be utilized to improve the accuracy of link prediction, provide better query results, rank web pages, and study information flow and spread in complex networks. In this paper, we model a social network as a directed graph $G$ , and consider the social hierarchy as DAG (directed acyclic graph) of $G$ , denoted as $G_D$ . By DAG, all the vertices in $G$ can be partitioned into different levels, the vertices at the same level represent a disjoint group in the social hierarchy, and all the edges in DAG follow one direction. The main issue we study in this paper is how to find DAG $G_D$ in $G$ . The approach we take is to find $G_D$ by removing all possible cycles from $G$ such that $G = {\cal U}(G) \cup G_D$ , where ${\cal U}(G)$ is a maximum Eulerian subgraph which contains all possible cycles. We give the reasons for doing so, investigate the properties of $G_D$ found, and discuss the applications. In addition, we develop a novel two-phase algorithm, called Greedy-&-Refine, which greedily computes an Eulerian subgraph and then refines this greedy solution to find the maximum Eulerian subgraph. We give a bound between the greedy solution and the optimal. The quality of our greedy approach is high. We conduct comprehensive experimental studies over 14 real-world datasets. The results show that our algorithms are at least two orders of magnitude faster than the baseline algorithm.

Snarks, Hypohamiltonian Graphs and Non-Supereulerian Graphs

A graph G is hypohamiltonian if it is not Hamiltonian but for each $$v\in V(G)$$vźV(G), the graph $$G-v$$G-v is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. A graph G is called collapsible if for every even subset $$R\subseteq V(G)$$R⊆V(G), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is reduced if it has no nontrivial collapsible subgraphs. In this note, we first prove that all hypohamiltonian cubic graphs are reduced non-supereulerian graphs. Then we introduce an operation to construct graphs from hypohamiltonian cubic graphs such that the resulting graphs are 3-edge-connected non-supereulerian reduced graphs and cannot be contracted to a snark. This disproves two conjectures, one of which was first posed by Catlin et al. in [Congr. Num. 76:173---181, 1990] and in [J. Combin. Theory, Ser B 66:123---139, 1996], and was posed again by Li et al. in [Acta Math. Sin. English Ser 30(2):291---304, 2014] and by Yang in [Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Universite Paris-Sub, September 27, 2013], respectively, the other one was posed by Yang 2013.

Supereulerian graphs with small circumference and 3-connected hamiltonian claw-free graphs

A graph G is supereulerian if it has a spanning eulerian subgraph. We prove that every 3-edge-connected graph with the circumference at most 11 has a spanning eulerian subgraph if and only if it is not contractible to the Petersen graph. As applications, we determine collections F1, F2 and F3 of graphs to prove each of the following(i) Every 3-connected {K1,3,Z9}-free graph is hamiltonian if and only if its closure is not a line graph L(G) for some G∈F1.(ii) Every 3-connected {K1,3,P12}-free graph is hamiltonian if and only if its closure is not a line graph L(G) for some G∈F2.(iii) Every 3-connected {K1,3,P13}-free graph is hamiltonian if and only if its closure is not a line graph L(G) for some G∈F3.

Supereulerian graphs with width [formula omitted] and [formula omitted]-collapsible graphs

For an integer s>0 and for u,v∈V(G) with u≠v, an (s;u,v)-trail-system of G is a subgraph H consisting of s edge-disjoint (u,v)-trails. A graph is supereulerian with widths if for any u,v∈V(G) with u≠v, G has a spanning (s;u,v)-trail-system. The supereulerian widthμ′(G) of a graph G is the largest integer s such that G is supereulerian with width k for every integer k with 0≤k≤s. Thus a graph G with μ′(G)≥2 has a spanning Eulerian subgraph. Catlin (1988) introduced collapsible graphs to study graphs with spanning Eulerian subgraphs, and showed that every collapsible graph G satisfies μ′(G)≥2 (Catlin, 1988; Lai et al., 2009). Graphs G with μ′(G)≥2 have also been investigated by Luo et al. (2006) as Eulerian-connected graphs. In this paper, we extend collapsible graphs to s-collapsible graphs and develop a new related reduction method to study μ′(G) for a graph G. In particular, we prove that K3,3 is the smallest 3-edge-connected graph with μ′<3. These results and the reduction method will be applied to determine a best possible degree condition for graphs with supereulerian width at least 3, which extends former results in Catlin (1988) and Lai (1988).

An improved approximation algorithm for the traveling salesman problem with relaxed triangle inequality

Given a complete edge-weighted graph G, we present a polynomial time algorithm to compute a degree-four-bounded spanning Eulerian subgraph of 2G that has at most 1.5 times the weight of an optimal TSP solution of G. Based on this algorithm and a novel use of orientations in graphs, we obtain a (3β/4+3β2/4)-approximation algorithm for TSP with β-relaxed triangle inequality (β-TSP), where β≥1. A graph G is an instance of β-TSP, if it is a complete graph with edge weights c:E(G)→Q≥0 that are restricted as follows. For each triple of vertices u,v,w∈V(G), c({u,v})≤β(c({u,w})+c({w,v})).

Spanning trees and spanning Eulerian subgraphs with small degrees

Liu and Xu (1998) and Ellingham, Nam and Voss (2002) independently showed that every k-edge-connected simple graph G has a spanning tree T such that for each vertex v, dT(v)≤⌈d(v)k⌉+2. In this paper we show that every k-edge-connected graph G has a spanning tree T such that for each vertex v, dT(v)≤⌈d(v)−2k⌉+2; also if G has k edge-disjoint spanning trees, then T can be found such that for each vertex v, dT(v)≤⌈d(v)−1k⌉+1. This result implies that every (r−1)-edge-connected r-regular graph (with r≥4) has a spanning Eulerian subgraph whose degrees lie in the set {2,4,6}; also reduces the edge-connectivity needed for some theorems due to Barát and Gerbner (2014) and Thomassen (2008, 2013). Moreover these bounds for finding spanning trees are sharp.

Fan-Type Conditions for Spanning Eulerian Subgraphs

For a graph $$G$$G, let $$\delta _F(G)=\min \{\max \{d(u), d(v)\} | \text{ for } \text{ any }~u, v\in V(G)\, \text{ with } \text{ distance }~2\}$$?F(G)=min{max{d(u),d(v)}|foranyu,v?V(G)withdistance2}. A graph is supereulerian if it has a spanning Eulerian subgraph. Let $$p>0$$p>0, $$g>2$$g>2 and $$\epsilon $$∈ be given nonnegative numbers. Let $$\mathcal{Q}$$Q be the family of non-supereulerian graphs with order at most $$5(p-2)$$5(p-2). In this paper, we prove that for a 3-edge-connected graph $$G$$G of order $$n$$n, if $$G$$G satisfies a Fan-type condition $$\delta _F(G)\ge \frac{n}{(g-2)p}-\epsilon $$?F(G)?n(g-2)p-∈ and $$n$$n is sufficiently large, then $$G$$G is supereulerian if and only if $$G$$G is not contractible to a graph in $$\mathcal{Q}$$Q. Results on best possible values of $$p$$p and $$\epsilon $$∈ for such graphs to either be supereulerian or be contractible to the Petersen graph are given.

Supereulerian graphs with small matching number and 2-connected hamiltonian claw-free graphs

Motivated by the Chinese Postman Problem, Boesch, Suffel, and Tindell [The spanning subgraphs of Eulerian graphs, J. Graph Theory 1 (1977), pp. 79–84] proposed the supereulerian graph problem which seeks the characterization of graphs with a spanning Eulerian subgraph. Pulleyblank [A note on graphs spanned by Eulerian graphs, J. Graph Theory 3 (1979), pp. 309–310] showed that the supereulerian problem, even within planar graphs, is NP-complete. In this paper, we settle an open problem raised by An and Xiong on characterization of supereulerian graphs with small matching numbers. A well-known theorem by Chvátal and Erdös [A note on Hamilton circuits, Discrete Math. 2 (1972), pp. 111–135] states that if G satisfies α(G)≤κ(G), then G is hamiltonian. Flandrin and Li in 1989 showed that every 3-connected claw-free graph G with α(G)≤2 κ(G) is hamiltonian. Our characterization is also applied to show that every 2-connected claw-free graph G with α(G)≤3 is hamiltonian, with only one well-characterized exceptional class.