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Characterization of Forbidden Subgraphs for the Existence of Even Factors in a Graph

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Characterization of Forbidden Subgraphs for the Existence of Even Factors in a Graph Discrete Applied Mathematics 223 (2017) 135–139 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Characterization of forbidden subgraphs for the existence of even factors in a graph Liming Xiong School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, PR China article info a b s t r a c t ∼ Article history: In 2008, it is shown that if H D K1;3 (the claw), then every H-free graph G has an even Received 25 October 2015 factor if and only if δ.G/ ≥ 2 and every odd branch-bond of G having an edge-branch. In Received in revised form 15 February 2017 this paper, we characterize all connected graphs H with order at least three such that every Accepted 16 February 2017 H-free graph has an even factor if and only if δ.G/ ≥ 2 and every odd branch-bond of G has Available online 9 March 2017 an edge-branch. ' 2017 Elsevier B.V. All rights reserved. Keywords: Even factor Forbidden graph Chair-free Claw-free Branch-bond 1. Introduction All graphs considered in this paper are simple graphs. For the notation or terminology not defined here, see [2]. We say that a graph G is H-free if G contains no induced subgraph isomorphic to H; H is also called a forbidden subgraph. We often require H to be connected; if H has order two, then H-free graph is a graph without any edge; these graphs are of little interest. Thus, we need H to have order at least three. Using a forbidden subgraph condition to characterize the graphs with some properties is a common one; while it often imposes some necessary conditions, for example, hamiltonian graphs with 2-connectivity, traceable graphs with connectivity, 2-factor graph with minimum degree at least two; see the following three theorems. Here a path Pk is a path of order k. Theorem 1 (Faudree and Gould, [3]). Suppose A is a connected graph of order at least 3. Then every 2-connected A-free graph is hamiltonian if and only if A D P3. Theorem 2 (Faudree and Gould, [3]). Suppose A is a connected graph of order at least 3. Then every connected A-free is traceable if and only if A D P3. Although a graph to be hamiltonian (must have a 2-factor, with exactly one component) seems to be stronger than it has a 2-factor (may not be hamiltonian), the following result shows the forbidden subgraph condition is the same (comparing it with Theorem 1). Theorem 3 (Aldred, Fujisawa and Saito, [1]). Suppose A is a connected graph of order at least 3. Then every connected graph with δ.G/ ≥ 2 contains 2-factor if and only if A D P3. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.dam.2017.02.015 0166-218X/' 2017 Elsevier B.V. All rights reserved. 136 L. Xiong / Discrete Applied Mathematics 223 (2017) 135–139 In [5], Lai considered the same problem in a slight different way. A graph is called supereulerian if it has a spanning eulerian subgraph. A graph H is called an induced minor if it is isomorphic to a contraction image of an induced subgraph of G.A wheel Wn is the graph obtained from an n-cycle Cn D u1u2 ::: unu1 and an additional vertex u by joining u with 0 ui for all i. Define the subdivided wheel Wn to be the graph obtained from Wn by subdividing each edge in E.Cn/ once. Let D f 0 V ≥ g Ω Wn n 2 . Note that 2-edge connectivity is a necessary condition for a graph to be supereulerian. Theorem 4 (Lai, [5]). Let G be a 2-edge connected graph. Then every 2-edge connected induced subgraph of G is supereulerian if, and only if, G has no induced minor isomorphic to a member in Ω. A graph is called trivial if it has only one vertex, nontrivial otherwise. A subgraph of a graph G is called even if its degrees are all even. An even subgraph H is called nontrivial if each of its components is nontrivial. A subgraph of G is called an even factor if it is an even nontrivial spanning subgraph of G. A supereulerian graph has an even factor with exactly one component. A nontrivial path is called a branch if it has only internal vertices of degree two and end vertices of degree not two. The length of a branch is the number of its edges. Note that a branch of length one has no internal vertex, is called an edge branch. We denote by B.G/ the set of branches of G. For any subset S of B.G/, we denote by G−S the subgraph obtained from G by deleting all edges and internal vertices of branches of S. A subset S of B.G/ is called a branch cut if G − S has more components than G. A minimal branch cut is called a branch-bond. A branch-bond is called odd if it has an odd number of branches. The idea of branch-bond comes originally from [9] which has also been applied in [4,8,7]. A star is the complete bipartite graph K1;m.A K1;3 is also called a claw. If G has an even factor, then δ.G/ ≥ 2 and G has no odd branch-bonds with a shortest branch of length at least 2. The following result shows that the two obvious necessary conditions for the existence of even factors are also sufficient for claw-free graphs. Note that there is a polynomial-time algorithm to determine whether a graph satisfies the hypothesis that every odd branch-bond has an edge branch, see, e.g., [6]. Theorem 5 (Xiong, [7]). Let G be a claw-free graph. Then G has an even factor if and only if δ.G/ ≥ 2 and every odd branch-bond of G has an edge branch. The results above motivate the research of extending Theorem 5. In this paper, we shall characterize all connected graphs H of order at least three guaranteeing that those obvious necessary conditions for the existence of even factors in G are also sufficient. 2. Main results and their proofs We start with the following auxiliary result. Theorem 6 (Xiong, [8]). Let ` be a positive integer. Then a graph G has a set of vertex-disjoint circuits containing all branches of length not ` if every odd branch-bond of G has a shortest branch of length `. The unique tree with a degree sequence 1,1,1,2,3 is called a chair.A circuit is a connected even graph with at least three vertices. In the proof of main theorem, we frequently take the symmetric difference of a graph and a cycle. Let H be a subgraph of a graph G, and let C be a cycle in G. Then we define H1C by H1C D V .H/ [ V .C/; E.H/1E.C/, where A1B denotes the symmetric difference of the sets A and B. Note that if H is an even graph, then H1C is also an even graph, but H1C may have more isolated vertices than H. We first prove the following result. 0 ≥ Theorem 7. Let G be either a P5-free graph or a chair-free graph other than W3. Then G has an even factor if and only if δ.G/ 2 and every odd branch-bond of G has an edge branch. Proof of Theorem 7. Necessity. An even factor should have minimum degree at least two and a graph has no even factor if it has an odd branch-bond which contains a shortest branch of length at least two. 0 ≥ Sufficiency. Let G be a P5-free or chair-free graph other than W3 with δ.G/ 2 in which every odd branch-bond has an edge branch. By Theorem 6, there exists an even subgraph C of G which contains all branches of length at least 2. Then C contains all the vertices of degree 2. Now choose an even subgraph C of G containing all the vertices of degree 2 so that C contains as many vertices of possible. We claim that C is a spanning subgraph of G. Assume V .C/ 6D V .G/. By the maximality of C, G n V .C/ does not contain an even subgraph. Let x 2 V .G/ n V .C/. By the choice of C, dG.x/ ≥ 3. Claim 1. NG.x/ is independent in G. Proof. Assume y1y2 2 E.G/ for some fy1; y2g ⊂ NG.x/. Let C1 D C∆.xy1y2x/. Then C1 is an even subgraph of G. Moreover, since x 62 V .C/, fxy1; xy2g \ E.C/ D;. Then fx; y1; y2g ⊂ V .C1/ and hence V .C1/ D V .C/ [ fxg. This contradicts the maximality of C. L. Xiong / Discrete Applied Mathematics 223 (2017) 135–139 137 Claim 2. Let y1 and y2 be distinct neighbors of x and let z 2 .NG.y1/ \ NG.y2// n fxg. Then (a) dC .z/ D 2 and fy1z; y2zg ⊂ E.C/, and 0 0 0 0 0 (b) for each y 2 NG.x/ (possibly y 2 fy1; y2g) and z 2 NG.y / n fxg, zz 62 E.G/. Proof. (a) Let C2 D C∆.xy1zy2x/. Then C2 is an even subgraph of G. Since fxy1; xy2g\ E.C/ D;, we have fx; y1; y2g ⊂ V .C2/.
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