Let K 3,3 be the 3-graph with 15 vertices {xi, yi : 1 ≤ i ≤ 3} and {zij : 1 ≤ i, j ≤ 3}, and 11 edges {x1, x2, x3}, {y1, y2, y3} and {{xi, yj , zij} : 1 ≤ i, j ≤ 3}. We show that for large n, the unique largest K 3,3-free 3-graph on n vertices is a balanced blow-up of the complete 3-graph on 5 vertices. Our proof uses the stability method and a result on lagrangians of intersecting families tha...

For a graph G and a set F of connected graphs, G is said be F-free if G does not contain any member of F as an induced subgraph. We let G3(F) denote the set of all 3-connected F-free graphs. This paper is concerned with sets F of connected graphs such that |F| = 3 and G3(F) is finite. Among other results, we show that for an integer m > 3 and a connected graph T of order greater than or equal t...

A matchstick graph is a crossing-free unit-distance in the plane. Harborth conjectured 1981 that maximum number of edges with $n$ vertices $\lfloor 3n-\sqrt{12n-3}\rfloor$. Using Euler formula and isoperimetric inequality, it can be shown has no more than $3n-\sqrt{2\pi\sqrt{3}\cdot n}+O(1)$ edges. We improve this upper bound to $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2\pi\...

A graph G is N2-locally connected if for every vertex v in G, the edges not incident with v but having at least one end adjacent to v in G induce a connected graph. In 1990, Ryjác̆ek conjectured that every 3-connected N2-locally connected claw-free graph is hamiltonian. This conjecture is proved in this note.

We show that a maximal triangle-free graph on n vertices with minimum degree δ contains an independent set of 3δ − n vertices which have identical neighborhoods. This yields a simple proof that if the binding number of a graph is at least 3/2 then it has a triangle. This was conjectured originally by Woodall. We consider finite undirected graphs on n vertices with minimum degree δ. A maximal tr...

Gluck [6] has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K5-minor freeness guarantees the stress freeness. More generally, we prove that every Kr+2-minor free graph is generically r-stress free for 1 ≤ r ≤ 4. (This assertion is false for r ≥ 6.) Some further extensions are discussed.

A graph is k-choosable if it can be colored whenever every vertex has a list of available colors of size at least k. It is a generalization of graph coloring where all vertices do not have the same available colors. We show that every triangle-free plane graph without 6-, 7-, and 8-cycles is 3-choosable.

It is known that every triangle-free (equivalently, of girth at least 4) circle graph is 5-colourable (Kostochka, 1988) and that there exist examples of these graphs which are not 4-colourable (Ageev, 1996). In this note we show that every circle graph of girth at least 5 is 2-degenerate and, consequently, not only 3-colourable but even 3-choosable.

Let G be a planar triangle-free graph and let C be a cycle in G of length at most 8. We characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of the whole graph.

A natural way of increasing our understanding NP-complete graph problems is to restrict the input a special class. Classes H-free graphs, that is, graphs do not contain some H as an induced subgraph, have proven be ideal testbed for such complexity study. However, if forbidden contains cycle or claw, then these often stay NP-complete. recent study (MFCS 2019) on k-Colouring problem shows we may...