Subgraphs of colour-critical graphs

Some problems and results on the distribution of subgraphs in colour-critical graphs are discussed. In section 3 arbitrarily largek-critical graphs withn vertices are constructed such that, in order to reduce the chromatic number tok−2, at leastc _k n ² edges must be removed. In section 4 it is proved that a 4-critical graph withn vertices contains at mostn triangles. Further it is proved that ak-critical graph which is not a complete graph contains a (k−1)-critical graph which is not a complete graph.     

Sparse color-critical graphs and hypergraphs with no short cycles

We give constructions of color-critical graphs and hypergraphs with no short cycles and with relatively few edges. In particular, we show that, for each n ≧ 3, the smallest number of edges in a 3-critical triangle-free n-graph (hypergraph) with m vertices is m + o(m) as m → ∞. Also, for each r ≧ 4, there exists an r-critical triangle-free 2-graph (graph) with m vertices and at most (r ? (7/3))m + o(m) edges. Weaker results are obtained for the existence of r-critical graphs containing no cycle of length at most / > 3.     

On the Number of Edges in Colour-Critical Graphs and Hypergraphs

A (hyper)graph G is called k-critical if it has chromatic number k, but every proper sub(hyper)graph of it is (k-1)-colourable. We prove that for sufficiently large k, every k-critical triangle-free graph on n vertices has at least (k-o(k))n edges. Furthermore, we show that every (k+1)-critical hypergraph on n vertices and without graph edges has at least (k-3/³?{k}) n(k-3/\sqrt[3]{k}) n edges. Both bounds differ from the best possible bounds by o(kn) even for graphs or hypergraphs of arbitrary girth.     

A characterization of diameter-2-critical graphs with no antihole of length four

A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n ²/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.     

Color-critical graphs have logarithmic circumference

A graph G is k-critical if every proper subgraph of G is (k−1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least , improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed . We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.     

Large faces in 4-critical planar graphs with minimum degree 4

We prove that the size of the largest face of a 4-critical planar graph with 4 is at most one half the number of its vertices. Letf(n) denote the maximum of the sizes of largest faces of all such graphs withn vertices (n sufficiently large). We present an infinite family of graphs that shows .All three authors gratefully acknowledge the support of the National Science and Engineering Research Council of Canada.     

Characterizing defect n-extendable graphs and -critical graphs

A near perfect matching is a matching saturating all but one vertex in a graph. Let G be a connected graph. If any n independent edges in G are contained in a near perfect matching where n is a positive integer and n(|V(G)|-2)/2, then G is said to be defect n-extendable. If deleting any k vertices in G where k|V(G)|-2, the remaining graph has a perfect matching, then G is a k-critical graph. This paper first shows that the connectivity of defect n-extendable graphs can be any integer. Then the characterizations of defect n-extendable graphs and (2k+1)-critical graphs using M-alternating paths are presented.     

On a conjecture of Murty and Simon on diameter 2-critical graphs

A graph G is diameter 2-critical if its diameter is two, and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order n is at most n²/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an association with total domination to prove the conjecture for the graphs whose complements have diameter three.     

Competition graphs and clique dimensions

Here it is proved that for almost all simple graphs over n vertices one needs Ω(n^4/3(log n)^?4/3) extra vertices to obtain them as a double competition graph of a digraph. on the other hand O(n^5/3) extra vertices are always sufficient. Several problems remain open.     

Small graphs with chromatic number 5: A computer search

In this article we give examples of a triangle-free graph on 22 vertices with chromatic number 5 and a K₄-free graph on 11 vertices with chromatic number 5. We very briefly describe the computer searches demonstrating that these are the smallest possible such graphs. All 5-critical graphs on 9 vertices are exhibited. © 1995 John Wiley & Sons, Inc.     

A proof of a conjecture on diameter 2-critical graphs whose complements are claw-free

A graph G is diameter 2-critical if its diameter is 2, and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order n is at most n²/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an important association with total domination to prove the conjecture for the graphs whose complements are claw-free.     

Total domination of graphs and small transversals of hypergraphs

The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvátal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.     

On the minimal number of edges in color-critical graphs

A graphG isk-critical if it has chromatic numberk, but every proper subgraph of it is (k?1)-colorable. This paper is devoted to investigating the following question: for givenk andn, what is the minimal number of edges in ak-critical graph onn vertices, with possibly some additional restrictions imposed? Our main result is that for everyk≥4 andn>k this number is at least $\left( {\frac{{k - 1}}{2} + \frac{{k - 3}}{{2(k^2 - 2k - 1)}}} \right)n$ , thus improving a result of Gallai from 1963. We discuss also the upper bounds on the minimal number of edges ink-critical graphs and provide some constructions of sparsek-critical graphs. A few applications of the results to Ramsey-type problems and problems about random graphs are described.     

The structure and maximum number of maximum independent sets in trees

A subset of vertices is a maximum independent set if no two of the vertices are joined by an edge and the subset has maximum cardinality. in this paper we answer a question posed by Herb Wilf. We show that the greatest number of maximum independent sets for a tree of n vertices is We give the families of trees on which these maxima are achieved. Proving which trees are extremal depends upon the structure of maximum independent sets in trees. This structure is described in terms of adjacency rules between three types of vertices, those which are in all, no, or some maximum independent sets. We show that vertices that are in some but not all maximum independent sets of the tree are joined in pairs by the α-critical edges (edges whose removal increases the size of a maximum independent set). The number of maximum independent sets is shown to depend on the structure within the tree of the α-critical edges.     

Local structure of 5- and 6-connected graphs

We prove that if graph on n vertices is minimally and contraction critically 5-connected, then it has 4n/7 vertices of degree 5. We also prove that if graph on n vertices is minimally and contraction critically 6-connected, then it has n/2 vertices of degree 6. Bibliography: 7 titles.     

On color critical graphs

It is shown that the minimal number of edges which have to be omitted from a (k + 1)-critical graph on n vertices in order to make it bipartite is at least ^k₂ for n large enough. This bound is best possible. Various related questions are considered.     

Paths with restricted degrees of their vertices in planar graphs

In this paper it is proved that every 3-connected planar graph contains a path on 3 vertices each of which is of degree at most 15 and a path on 4 vertices each of which has degree at most 23. Analogous results are stated for 3-connected planar graphs of minimum degree 4 and 5. Moreover, for every pair of integers n 3, k 4 there is a 2-connected planar graph such that every path on n vertices in it has a vertex of degree k.     

Self‐destructive percolation as a limit of forest‐fire models on regular rooted trees

Let T be a regular rooted tree. For every natural number n, let T_n be the finite subtree of vertices with graph distance at most n from the root. Consider the following forest‐fire model on T_n: Each vertex can be “vacant” or “occupied”. At time 0 all vertices are vacant. Then the process is governed by two opposing mechanisms: Vertices become occupied at rate 1, independently for all vertices. Independently thereof and independently for all vertices, “lightning” hits vertices at rate λ(n) > 0. When a vertex is hit by lightning, its occupied cluster becomes vacant instantaneously. Now suppose that λ(n) decays exponentially in n but much more slowly than 1/|T_n|, where |T_n| denotes the number of vertices of T_n. We show that then there exist such that between time 0 and time the forest‐fire model on T_n tends to the following process on T as n goes to infinity: At time 0 all vertices are vacant. Between time 0 and time τ vertices become occupied at rate 1, independently for all vertices. Immediately before time τ there are infinitely many infinite occupied clusters. At time τ all these clusters become vacant. Between time τ and time vertices again become occupied at rate 1, independently for all vertices. At time all occupied clusters are finite. This process is a dynamic version of self‐destructive percolation. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 86–113, 2017     

An upper bound for the path number of a graph

The path number of a graph G, denoted p(G), is the minimum number of edge-disjoint paths covering the edges of G. Lovász has proved that if G has u odd vertices and g even vertices, then p(G) ≤ 1/2 u + g - 1 ≤ n - 1, where n is the total number of vertices of G. This paper clears up an error in Lovász's proof of the above result and uses an extension of his construction to show that p(G) ≤ 1/2 u + [3/4g] ≤ [3/4n].     

A tight upper bound on the cover time for random walks on graphs

We prove that the expected time for a random walk to visit all n vertices of a connected graph is at most 4/27n³ + o(n³).