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1.
    
We prove that a graph G contains no induced ‐vertex path and no induced complement of a ‐vertex path if and only if G is obtained from 5‐cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement, where split unification is a new class‐preserving operation introduced here.  相似文献   

2.
    
We show that the minimum set of unordered graphs that must be forbidden to get the same graph class characterized by forbidding a single ordered graph is infinite. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 71–76, 1999  相似文献   

3.
    
It is proven that if G is a 3‐connected claw‐free graph which is also H1‐free (where H1 consists of two disjoint triangles connected by an edge), then G is hamiltonian‐connected. Also, examples will be described that determine a finite family of graphs such that if a 3‐connected graph being claw‐free and L‐free implies G is hamiltonian‐connected, then L . © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 104–119, 2002  相似文献   

4.
    
Let H be a given graph. A graph G is said to be H‐free if G contains no induced copies of H. For a class of graphs, the graph G is ‐free if G is H‐free for every . Bedrossian characterized all the pairs of connected subgraphs such that every 2‐connected ‐free graph is hamiltonian. Faudree and Gould extended Bedrossian's result by proving the necessity part of the result based on infinite families of non‐hamiltonian graphs. In this article, we characterize all pairs of (not necessarily connected) graphs such that there exists an integer n0 such that every 2‐connected ‐free graph of order at least n0 is hamiltonian.  相似文献   

5.
    
A graph is H‐free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique‐width. Brandstädt, Le, and Mosca erroneously claimed that the gem and co‐gem are the only two 1‐vertex P4‐extensions H for which the class of H‐free chordal graphs has bounded clique‐width. In fact we prove that bull‐free chordal and co‐chair‐free chordal graphs have clique‐width at most 3 and 4, respectively. In particular, we find four new classes of H‐free chordal graphs of bounded clique‐width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H‐free chordal graphs has bounded clique‐width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of ‐free graphs has bounded clique‐width via a reduction to K4‐free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique‐width of H‐free weakly chordal graphs.  相似文献   

6.
    
We study the following question: How few edges can we delete from any H $H$-free graph on n $n$ vertices to make the resulting graph k $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For H $H$ any fixed odd cycle, we determine the answer up to a constant factor when n $n$ is sufficiently large. We also prove an upper bound when H $H$ is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges.  相似文献   

7.
All bipartite graphs whose third largest Laplacian eigenvalue is less than 3 have been characterized by Zhang. In this paper, all connected non-bipartite graphs with third largest Laplacian eigenvalue less than three are determined.  相似文献   

8.
    
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9.
The definition of the ascending suhgraph decomposition was given by Alavi. It has been conjectured that every graph of positive size has an ascending subgraph decomposition. In this paper it is proved that the regular graphs under some conditions do have an ascending subgraph decomposition.  相似文献   

10.
    
In this note we strengthen the stability theorem of Erd?s and Simonovits. Write Kr(s1, …, sr) for the complete r‐partite graph with classes of sizes s1, …, sr and Tr(n) for the r‐partite Turán graph of order n. Our main result is: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with e(G)>(1?1/r?ε)n2/2 satisfies one of the conditions:
  • (a) G contains a $K_{r+1} (lfloor c,mbox{ln},n rfloor,ldots,lfloor c,mbox{ln},n rfloor,lceil n^{{1}-sqrt{c}}rceil)In this note we strengthen the stability theorem of Erd?s and Simonovits. Write Kr(s1, …, sr) for the complete r‐partite graph with classes of sizes s1, …, sr and Tr(n) for the r‐partite Turán graph of order n. Our main result is: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with e(G)>(1?1/r?ε)n2/2 satisfies one of the conditions:
    • (a) G contains a $K_{r+1} (lfloor c,mbox{ln},n rfloor,ldots,lfloor c,mbox{ln},n rfloor,lceil n^{{1}-sqrt{c}}rceil)$;
    • (b) G differs from Tr(n) in fewer than (ε1/3+c1/(3r+3))n2 edges.
    Letting µ(G) be the spectral radius of G, we prove also a spectral stability theorem: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with µ(G)>(1?1/r?ε)n satisfies one of the conditions:
    • (a) G contains a $K_{r+1}(lfloor c,mbox{ln},nrfloor,ldots,lfloor c,mbox{ln},nrfloor,lceil n^{1-sqrt{c}}rceil)$;
    • (b) G differs from Tr(n) in fewer than (ε1/4+c1/(8r+8))n2 edges.
    © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 362–368, 2009  相似文献   

11.
    
Given a family and a host graph H, a graph is ‐saturated relative to H if no subgraph of G lies in but adding any edge from to G creates such a subgraph. In the ‐saturation game on H, players Max and Min alternately add edges of H to G, avoiding subgraphs in , until G becomes ‐saturated relative to H. They aim to maximize or minimize the length of the game, respectively; denotes the length under optimal play (when Max starts). Let denote the family of odd cycles and the family of n‐vertex trees, and write F for when . Our results include , for , for , and for . We also determine ; with , it is n when n is even, m when n is odd and m is even, and when is odd. Finally, we prove the lower bound . The results are very similar when Min plays first, except for the P4‐saturation game on .  相似文献   

12.
    
Let be a family of connected graphs. A graph G is said to be ‐free if G is H‐free for every graph H in . We study the relation between forbidden subgraphs in a connected graph G and the resulting toughness of G. In particular, we consider the problem of characterizing the graph families such that every large enough connected ‐free graph is t‐tough. In this article, we solve this problem for every real positive number t. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 191–202, 2013  相似文献   

13.
    
For an oriented graph G $$ G $$ , let β ( G ) $$ beta (G) $$ denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any m $$ m $$ -edge oriented graph G $$ G $$ satisfies β ( G ) = m / 2 − Ω ( m 3 / 4 ) $$ beta (G)=m/2-Omega left({m}^{3/4}right) $$ . We observe that if an oriented graph G $$ G $$ has a fixed forbidden subgraph B $$ B $$ , the bound β ( G ) = m / 2 − Ω ( m 3 / 4 ) $$ beta (G)=m/2-Omega left({m}^{3/4}right) $$ is sharp as a function of m $$ m $$ if B $$ B $$ is not bipartite, but the exponent 3 / 4 $$ 3/4 $$ in the lower order term can be improved if B $$ B $$ is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in [ 3 / 4 , 1 ] $$ left[3/4,1right] $$ is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear-time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets.  相似文献   

14.
    
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15.
    
Broersma and Veldman proved that every 2-connected claw-free and P 6-free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected claw-heavy and P 6-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and P 6-o-heavy but not hamiltonian. In this paper, we further give some Ore-type degree conditions restricting to induced copies of P 6 of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.  相似文献   

16.
    
Let be a set of connected graphs, each of which has order at least three, and suppose that there exist infinitely many connected -free graphs of minimum degree at least  two and all except for finitely many of them have a 2-factor. In [J. Graph Theory, 64 (2010), 250–266], we proved that if , then one of the members in is a star. In this article, we determine the remaining members of and hence give a complete characterization of the pairs and triples of forbidden subgraphs.  相似文献   

17.
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18.
    
We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.  相似文献   

19.
    
For a positive integer k, a k‐coloring of a graph is a mapping such that whenever . The Coloring problem is to decide, for a given G and k, whether a k‐coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k‐Coloring instead. We survey known results on the computational complexity of Coloring and k‐Coloring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.  相似文献   

20.
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