Triangle-free graphs with no six-vertex induced path

The graphs with no five-vertex induced path are still not understood. But in the triangle-free case, we can do this and one better; we give an explicit construction for all triangle-free graphs with no six-vertex induced path. Here are three examples: the 16-vertex Clebsch graph, the graph obtained from an 8-cycle by making opposite vertices adjacent, and the graph obtained from a complete bipartite graph by subdividing a perfect matching. We show that every connected triangle-free graph with no six-vertex induced path is an induced subgraph of one of these three (modulo some twinning and duplication).     

Acyclic edge colorings of planar graphs and seriesparallel graphs

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1. This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).     

A note on convex renorming and fragmentability

Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly locally uniformly rotund, then its weak topology is fragmented by a metric which is stronger than the norm topology.     

Claw-free graphs. III. Circular interval graphs

Construct a graph as follows. Take a circle, and a collection of intervals from it, no three of which have union the entire circle; take a finite set of points V from the circle; and make a graph with vertex set V in which two vertices are adjacent if they both belong to one of the intervals. Such graphs are “long circular interval graphs,” and they form an important subclass of the class of all claw-free graphs. In this paper we characterize them by excluded induced subgraphs. This is a step towards the main goal of this series, to find a structural characterization of all claw-free graphs.This paper also gives an analysis of the connected claw-free graphs G with a clique the deletion of which disconnects G into two parts both with at least two vertices.     

Maximum bipartite subgraphs of cubic triangle-free planar graphs

Thomassen recently proved, using the Tutte cycle technique, that if G is a 3-connected cubic triangle-free planar graph then G contains a bipartite subgraph with at least edges, improving the previously known lower bound . We extend Thomassen’s technique and further improve this lower bound to .     

Boundaries of compact convex sets and fragmentability

If X is a compact convex set in a real locally convex space, B⊂X is said to be its boundary if every affine continuous function on X attains its maximum at some point of B. We study relations between fragmentability of B and the whole set X. As a byproduct we obtain a characterization of separable Asplund spaces. We also study the possibility of finding the Haar system in a boundary of a metrizable compact convex set.     

Light subgraphs in planar graphs of minimum degree 4 and edge‐degree 9

A graph H is light in a given class of graphs if there is a constant w such that every graph of the class which has a subgraph isomorphic to H also has a subgraph isomorphic to H whose sum of degrees in G is ≤ w. Let be the class of simple planar graphs of minimum degree ≥ 4 in which no two vertices of degree 4 are adjacent. We denote the minimum such w by w(H). It is proved that the cycle C_s is light if and only if 3 ≤ s ≤ 6, where w(C₃) = 21 and w(C₄) ≤ 35. The 4‐cycle with one diagonal is not light in , but it is light in the subclass consisting of all triangulations. The star K_1,s is light if and only if s ≤ 4. In particular, w(K_1,3) = 23. The paths P_s are light for 1 ≤ s ≤ 6, and heavy for s ≥ 8. Moreover, w(P₃) = 17 and w(P₄) = 23. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 261–295, 2003     

Reconstructing graphs from size and degree properties of their induced k-subgraphs

As a variant of the famous graph reconstruction problem we characterize classes of graphs of order n such that all their induced subgraphs on k?n vertices satisfy some property related to the number of edges or to the vertex degrees.We give complete solutions for the properties (i) to be regular, (ii) to be regular modulo m?2 or (iii) to have one of two possible numbers of edges. Furthermore, for an order n large enough, we give solutions for the properties (iv) to be bi-regular or (v) to have a bounded difference between the maximum and the minimum degree.     

Planarity of iterated line graphs

The line index of a graph G is the smallest k such that the kth iterated line graph of G is nonplanar. We show that the line index of a graph is either infinite or it is at most 4. Moreover, we give a full characterization of all graphs with respect to their line index.     

A characterization of level planar graphs

We present a characterization of level planar graphs in terms of minimal forbidden subgraphs called minimal level non-planar (MLNP) subgraph patterns. We show that an MLNP subgraph pattern is completely characterized by either a tree, a level non-planar cycle or a level planar cycle with certain path augmentations.     

Coloring some classes of mixed graphs

We consider the coloring problem for mixed graphs, that is, for graphs containing edges and arcs. A mixed coloring c is a coloring such that for every edge [x_i,x_j], c(x_i)≠c(x_j) and for every arc (x_p,x_q), c(x_p)<c(x_q). We will analyse the complexity status of this problem for some special classes of graphs.     

Construction of a family of graphs with a small induced proper subgraph with minimum degree 3

We investigate the following question proposed by Erd?s: Is there a constant c such that, for each n, if G is a graph with n vertices, 2n-1edges, andδ(G)?3, then G contains an induced proper subgraph H with at least cn vertices andδ(H)?3?Previously we showed that there exists no such constant c by constructing a family of graphs whose induced proper subgraph with minimum degree 3 contains at most vertices. In this paper we present a construction of a family of graphs whose largest induced proper subgraph with minimum degree 3 is K₄. Also a similar construction of a graph with n vertices and αn+β edges is given.     

Maximum induced matching problem on hhd-free graphs

We show that the maximum induced matching problem can be solved on hhd-free graphs in O(m²) time; hhd-free graphs generalize chordal graphs and the previous best bound was O(m³). Then, we consider a technique used by Brandstädt and Hoàng (2008) [4] to solve the problem on chordal graphs. Extending this, we show that for a subclass of hhd-free graphs that is more general than chordal graphs the problem can be solved in linear time. We also present examples to demonstrate the tightness of our results.     

Unavoidable doubly connected large graphs

A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs:

(1) P_n, a path on n vertices;

(2) K_1,n^s, the graph obtained from K_1,n by subdividing an edge once;

(3) K_2,ne, the graph obtained from K_2,n by deleting an edge;

(4) K_2,n⁺, the graph obtained from K_2,n by adding an edge between the two degree-n vertices x₁ and x₂, and a pendent edge at each x_i.

Two applications of this result are also discussed in the paper.     

Uniformly optimal graphs in some classes of graphs with node failures

The uniformly optimal graph problem with node failures consists of finding the most reliable graph in the class Ω(n,m) of all graphs with n nodes and m edges in which nodes fail independently and edges never fail. The graph G is called uniformly optimal in Ω(n,m) if, for all node-failure probabilities q∈(0,1), the graph G is the most reliable graph in the class of graphs Ω(n,m). This paper proves that the multipartite graphs K(b,b+1,…,b+1,b+2) are uniformly optimal in their classes Ω((k+2)(b+1),(k²+3k+2)(b+1)²/2−1), where k is the number of partite sets of size (b+1), while for i>2, the multipartite graphs K(b,b+1,…,b+1,b+i) are not uniformly optimal in their classes Ω((k+2)b+k+i,(k+2)(k+1)b²/2+(k+1)(k+i)b+k(k+2i−1)/2).