On graphs with no induced subdivision of K4 (original) (raw)
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P_4-Colorings and P_4-Bipartite Graphs
Discrete Mathematics & Theoretical Computer Science
A vertex partition of a graph into disjoint subsets V_is is said to be a P_4-free coloring if each color class V_i induces a subgraph without chordless path on four vertices (denoted by P_4). Examples of P_4-free 2-colorable graphs (also called P_4-bipartite graphs) include parity graphs and graphs with ''few'' P_4s like P_4-reducible and P_4-sparse graphs. We prove that, given k≥ 2, \emphP_4-Free k-Colorability is NP-complete even for comparability graphs, and for P_5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P_4-bipartite graphs with special P_4-structure. In particular, we show that the SPGC is true for P_4-bipartite graphs with one P_3-free color class meeting every P_4 at a midpoint.
Triangle-free partial graphs and edge covering theorems
Discrete Mathematics, 1982
In Section 1 some lower bounds are given for the maximal number of edges of a (p-l)colorable partial graph. Among others we show that a graph on n vertices with m edges has a (p-l)-colorable partiai graph with at least mT,.$(;) edges, where T,,p denotes the so called Turin number. These results are used to obtain upper bounds for special edge covering numbers of graphs. In Section 2 we prove the following theorem: If G is a simple graph and or. is the maximal cardinality of a triangle-free edge set of G, then the edges of G can be covered by p triangles and edges. In Section 3 related questions are examined. We will consider only loopless graphs withlout multiple edg.zs. If G = (X, E) is a graph, then the edge set Fc E together with the spanned vertices define a partial graph of G which will be denoted by the same letter F for simplicity reasons. KP stands for a p-clique (compiete graph on p vertices); K3 will be called a
Discrete Applied Mathematics, 2009
A circular-arc graph is the intersection graph of a family of arcs on a circle. A characterization by forbidden induced subgraphs for this class of graphs is not known, and in this work we present a partial result in this direction. We characterize circular-arc graphs by a list of minimal forbidden induced subgraphs when the graph belongs to any of the following classes: P 4 -free graphs, paw-free graphs, claw-free chordal graphs and diamond-free graphs.
On Tucker's proof of the strong perfect graph conjecture for -free graphs
Discrete Mathematics, 2001
In this note, the authors generalize the ideas presented by Tucker in his proof of the Strong Perfect Graph Conjecture for (K4 − e)-free graphs in order to ÿnd a vertex v in G whose special neighborhood allows to extend a !(G)-vertex coloring of G − v to a !(G)-vertex coloring of G.
A characterization of K2,4K_{2,4}K2,4-minor-free graphs
We provide a complete structural characterization of K2,4K_{2,4}K2,4-minor-free graphs. The 333-connected K2,4K_{2,4}K2,4-minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains K4K_4K4 and, for each nge5n \ge 5nge5, 2n−82n-82n−8 nonisomorphic graphs of order nnn. To describe the 222-connected K2,4K_{2,4}K2,4-minor-free graphs we use xyxyxy-outerplanar graphs, graphs embeddable in the plane with a Hamilton xyxyxy-path so that all other edges lie on one side of this path. We show that, subject to an appropriate connectivity condition, xyxyxy-outerplanar graphs are precisely the graphs that have no rooted K2,2K_{2,2}K2,2-minor where xxx and yyy correspond to the two vertices on one side of the bipartition of K2,2K_{2,2}K2,2. Each 222-connected K2,4K_{2,4}K2,4-minor-free graph is then (i) outerplanar, (ii) the union of three xyxyxy-outerplanar graphs and possibly the edge xyxyxy, or (iii) obtained from a 333-connected K2,4K_{2,4}K2,4-minor-free graph by replacing each edge $x_iy_...
A note on Brooks' theorem for triangle-free graphs
For the class of triangle-free graphs Brooks' Theorem can be restated in terms of forbidden induced subgraphs, i.e. let G be a triangle-free and K 1,r+1-free graph. Then G is r-colourable unless G is isomorphic to an odd cycle or a complete graph with at most two vertices. In this note we present an improvement of Brooks' Theorem for triangle-free and rsunshade-free graphs. Here, an r-sunshade (with r ≥ 3) is a star K 1,r with one branch subdivided. A classical result in graph colouring theory is the theorem of Brooks [2], asserting that every graph G is (∆(G))-colourable unless G is isomorphic to an odd cycle or a complete graph. Bryant [3] simplified this proof with the following characterization of cycles and complete graphs. Thereby he highlights the exceptional role of the cycles and complete graphs in Brooks' Theorem. Here we give a new elementary proof of this characterization. Proposition 1 (Bryant [3]). Let G be a 2-connected graph. Then G is a cycle or a complete graph if and only if G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Proof. Let G be a 2-connected graph of order n. If G is a cycle or a complete graph, then obviously G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Hence, assume that G is neither a cycle nor a complete graph and that G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Note that then there exists at least one vertex v of G with 2 < d G (v) < n − 1. Since G
A Note on the Inducibility of 4$$ 4 -Vertex Graphs
Graphs and Combinatorics, 2014
There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph H is its extremal density, as an induced subgraph of G, where |G| → ∞. Already for 4-vertex graphs many questions are still open. Thus, the inducibility of the 4-path was addressed in a construction of Exoo (1986), but remains unknown. Refuting a conjecture of Erdős, Thomason (1997) constructed graphs with a small density of both 4-cliques and 4-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.
Graphs with no induced C4 and 2K2
Discrete Mathematics, 1993
We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class G of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members of G, the number of cliques (as well as the number of maximal independent sets) cannot exceed the number of vertices. Moreover, these graphs are almost extremal to the theorem of .
On the cyclic decomposition of complete graphs into bipartite graphs
2001
Let G be a graph with n edges. It is known that there exists a cyelic Gdecomposition of K 2n+1 if and only if G has a p-Iabeling. An a-labeling of G easily yields both a cyelic G-decomposition of Kn,n and of K2nx+l for all positive integers x. It is well-known that certain classes of bipartite graphs (including certain trees) do not have a-Iabelings. :Moreover, there are bipartite graphs with n edges which do not cyclically divide Kn,n. In this article, \ve introduce the concept of an ordered p-labeling (denoted by p+) of a bipartite graph, and prove that if a graph G with n edges has a p+ -labeling, then there is a cyclic G-decomposition of K 2nx+1 for all positive integers .17. vVe also introduce the concept of a O-labeling which is a more restrictive p+ -labeling. We conjecture that all forests have a p+labeling and show that the vertex-disjoint union of any finite collection of graphs that admit a-Iabelings admits a O-labeling.