Supereulerian Graphs Research Articles

Forbidden Pairs for Connected Even Factors in Supereulerian Graphs

Forbidden Pairs for Connected Even Factors in Supereulerian Graphs

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)

Strengthened Ore conditions for (s,t)-supereulerian graphs

For integers s ≥ 0 and t ≥ 0 , a graph G is ( s , t ) -supereulerian if for any disjoint edge sets X , Y ⊆ E ( G ) with | X | ≤ s and | Y | ≤ t , G has a spanning closed trail that contains X and avoids Y . Pulleyblank (1979) 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 (Catlin, 1988) showed that every simple graph G on n vertices with δ ( G ) ≥ n 5 − 1 , when n is sufficiently large, is ( 0 , 0 ) -supereulerian or is contractible to K 2 , 3 . A function j 0 ( s , t ) has been found that every ( s , t ) -supereulerian graph must have edge connectivity at least j 0 ( s , t ) . For any nonnegative integers s and t , we obtain best possible Ore conditions to assure a simple graph on n vertices to be ( s , t ) -supereulerian as stated in the following. (i) For any real numbers α and β with 0 < α < 1 , there exists a family of finitely many graphs F ( α , β ; s , t ) such that if κ ′ ( G ) ≥ j 0 ( s , t ) and if for any nonadjacent vertices u , v ∈ V ( G ) , d G ( u ) + d G ( v ) ≥ α n + β , then either G is ( s , t ) -supereulerian, or G is contractible to a member in F ( α , β ; s , t ) . (ii) If κ ′ ( G ) ≥ j 0 ( s , t ) and if for any nonadjacent vertices u , v ∈ V ( G ) , d G ( u ) + d G ( v ) ≥ n − 1 , then when n is sufficiently large, either G is ( s , t ) -supereulerian, or G is contractible to one of the six well specified graphs. (iii) Suppose that δ ( G ) ≥ 5 . If (1) for any vertices u , v , w ∈ V ( G ) with E ( G [ { u , v , w } ] ) = 0̸ , d G ( u ) + d G ( v ) + d G ( w ) > n − 3 . then G is ( s , t ) -supereulerian if and only if κ ′ ( G ) ≥ j 0 ( s , t ) .

Polynomially determine if a graph is (s,3)-supereulerian

For integers s≥0 and t≥0, a graph G is (s,t)-supereulerian if for any disjoint edge sets X,Y⊆E(G) with |X|≤s and |Y|≤t, G has a spanning closed trail that contains X and avoids Y. Pulleyblank in 1979 showed that determining whether a graph is (0,0)-supereulerian, even when restricted to planar graphs, is NP-complete. We investigate the value of the smallest integer j(s,t) such that every j(s,t)-edge-connected graph is (s,t)-supereulerian, and show thatj(s,t)={max⁡{4,t+2} if 0≤s≤1, or (s,t)∈{(2,0),(2,1),(3,0)},5 if (s,t)∈{(2,2),(3,1)},s+t+1−(−1)s2 if s≥2 and s+t≥5. As applications, we obtain a characterization of (s,t)-supereulerian graphs when t≥3 in terms of edge-connectivities, and show that when t≥3, there exists a polynomial time algorithm to determine if a graph is (s,t)-supereulerian.

Spectral and extremal conditions for supereulerian graphs

A graph is supereulerian if it contains a spanning closed trail. We prove several Erds-type extremal size conditions with a lower bounded minimum degree for a graph to be supereulerian and with different edge-connectivity, with the corresponding extremal graphs characterized. These results are then applied to prove sufficient conditions involving adjacency spectral radius and signless Laplacian spectral radius for a graph to be supereulerian.

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 characterization of graphs with supereulerian line graphs

The line graph of a graph G is a simple graph with being its vertex set, where two vertices are adjacent in whenever the corresponding edges share a common vertex in G. A graph H is even if every vertex of H has even degree, and a graph is supereulerian if it has a spanning closed trail. We obtain a characterization for a graph G to have a supereulerian line graph , as follows: for a connected graph G with , the line graph has a spanning closed trail if and only if G has an even subgraph H (possibly null) such that both G remains connected after deleting all degree 2 vertices not in H, and every degree 2 vertex not in H must be adjacent only to vertices of degree at least 3 in G.

Collapsible subgraphs of a 4-edge-connected graph

Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, graphs that have spanning eulerian subgraphs. Catlin in 1988 sharpened Jaeger’s result by showing that every 4-edge-connected graph is collapsible, graphs that are contractible configurations of supereulerian graphs. To further study collapsible subgraphs of a 4-edge-connected graph, in Catlin et al. (2009), it is shown that every 4-edge-connected graph remains collapsible after removing any two edges. We prove the following. •[(i)] Every 4-edge-connected G contains two vertices x,y such that one of x and y has minimum degree in G and both G−x and G−y are collapsible.•[(ii)] Let G be a 4-edge-connected graph and let X⊂E(G) be an edge subset with |X|≤3. Then G−X is collapsible if and only if X is not contained in a 4-edge-cut of G.•[(iii)] Let G be a 4-edge-connected graph and let X⊂E(G) be an edge subset with |X|≤4. Then G−X is collapsible if and only if G−X is not contractible to a member in {K2c,K2,K2,2,K2,3,K2,4}. These extend former results of Jaeger (1979) and Catlin (1988).

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.

Pairs of forbidden subgraphs and 2-connected supereulerian graphs

Let G be a 2-connected claw-free graph. We show that •if G is N1,1,4-free or N1,2,2-free or Z5-free or P8-free, respectively, then G has a spanning Eulerian subgraph (i.e. a spanning connected even subgraph) or its closure is the line graph of a graph in a family of well-defined graphs,•if the minimum degree δ(G)≥3 and G is N2,2,5-free or Z9-free, respectively, then G has a spanning Eulerian subgraph or its closure is the line graph of a graph in a family of well-defined graphs.Here Zi (Ni,j,k) denotes the graph obtained by attaching a path of length i≥1 (three vertex-disjoint paths of lengths i,j,k≥1, respectively) to a triangle.Combining our results with a result in [Xiong (2014)], we prove that all 2-connected hourglass-free claw-free graphs G with one of the same forbidden subgraphs above (or additionally δ(G)≥3) are hamiltonian with the same excluded families of graphs. In particular, we prove that every 3-edge-connected claw-free hourglass-free graph that is N2,2,5-free or Z9-free is hamiltonian.

Forbidden set of induced subgraphs for [formula omitted]-connected supereulerian graphs

Let H={H1,…,Hk} be a set of connected graphs. A graph is said to be H-free if it does not contain any member of H as an induced subgraph. We show that if the following statements hold, •|H|≥2,•K1,3∈H,•for any integer n0, every 2-connected H-free graph G of order at least n0 is supereulerian, i.e.,G has a spanning closed trail,then H∖{K1,3} contains an Ni,j,k or a path where Ni,j,k denotes the graph obtained by attaching three vertex-disjoint paths of lengths i,j,k≥0 to a triangle.As an application, we characterize all the forbidden triples H with K1,3∈H such that every 2-connected H-free graph is supereulerian. As a byproduct, we also characterize minimal 2-connected non-supereulerian claw-free graphs.

Supereulerian width of dense graphs

For a graph G, the supereulerian widthμ′(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 (1988) proved that if a simple graph G on n≥17 vertices satisfy δ(G)≥n4−1, then μ′(G)≥2. In this paper, we show that for any real numbers a,b with 0<a<1 and any integer s>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=14,b=−32, we show that if n is sufficiently large, K3,3 is the only obstacle for a 3-edge-connected graph G to satisfy μ′(G)≥3.

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.

Lai's Conditions for Spanning and Dominating Closed Trails

A graph is supereulerian if it has a spanning closed trail. For an integer $r$, let ${\cal Q}_0(r)$ be the family of 3-edge-connected nonsupereulerian graphs of order at most $r$. For a graph $G$, define $\delta_L(G)=\min\{\max\{d(u), d(v) \}| \ \mbox{ for any $uv\in E(G)$} \}$. For a given integer $p\ge 2$ and a given real number $\epsilon$, a graph $G$ of order $n$ is said to satisfy a Lai's condition if $\delta_L(G)\ge \frac{n}{p}-\epsilon$. In this paper, we show that if $G$ is a 3-edge-connected graph of order $n$ with $\delta_L(G)\ge \frac{n}{p}-\epsilon$, then there is an integer $N(p, \epsilon)$ such that when $n&gt; N(p,\epsilon)$, $G$ is supereulerian if and only if $G$ is not a graph obtained from a graph $G_p$ in the finite family ${\cal Q}_0(3p-5)$ by replacing some vertices in $G_p$ with nontrivial graphs. Results on the best possible Lai's conditions for Hamiltonian line graphs of 3-edge-connected graphs or 3-edge-connected supereulerian graphs are given, which are improvements of the results in [J. Graph Theory 42(2003) 308-319] and in [Discrete Mathematics, 310(2010) 2455-2459].

Sufficient Conditions for a Digraph to be Supereulerian

AbstractA (di)graph is supereulerian if it contains a spanning eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we analyze a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the first author: if the arc‐connectivity of a digraph is not smaller than its independence number, then the digraph is supereulerian. As a support for this conjecture we prove it for digraphs that are semicomplete multipartite or quasitransitive and verify the analogous statement for undirected graphs.

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.

Supereulerian graphs and the Petersen graph

A graph G is supereulerian if G has a spanning eulerian subgraph. Boesch et al. [J. Graph Theory, 1, 79–84 (1977)] proposed the problem of characterizing supereulerian graphs. In this paper, we prove that any 3-edge-connected graph with at most 11 edge-cuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen graph. This extends a former result of Catlin and Lai [J. Combin. Theory, Ser. B, 66, 123–139 (1996)].

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 > 0 and for $${u, v \in V(G)}$$ 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 $${u, v \in V(G)}$$ with u ? v, G has a spanning (s; u, v)-path-system. The spanning connectivity ?*(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 $${u, v \in V(G)}$$ with u ? v. An edge counter-part of ?*(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 ?*(L(G)) ? 2 if and only if ?(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 ?*(L(G)) ? k if and only if ?(L(G)) ? max{3, k}.