Given a graph H, we say a graph G is H-saturated if G does not contain H as a subgraph and the addition of any edge e′ 6∈ E(G) results in H as a subgraph. In this paper, we construct (K4 − e)-saturated graphs with |E(G)| either the size of a complete bipartite graph, a 3-partite graph, or in the interval [ 2n− 4, ⌊ n 2 ⌋ ⌈ n 2 ⌉ − n+ 6 ] . We then extend the (K4−e)-saturated graphs to (Kt − e)-...

We give a computer-assisted proof of the fact that R(K5 − P3,K5) = 25. This solves one of the three remaining open cases in Hendry’s table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no (K5−P3,K5)-good graphs containing a K4 on 23 or 24 vertices, where a graph F is (G, H)-good if F does not contain G and the complement of F does not containH. The...

Arising from complete Cayley graphs Γn of odd cyclic groups Zn, an asymptotic approach is presented on connected labeled graphs whose vertices are labeled via equallymulticolored copies of K4 in Γn with adjacency of any two such vertices whenever they are represented by copies of K4 in Γn sharing two equally-multicolored triangles. In fact, these connected labeled graphs are shown to form a fam...

Graphs of treewidth at most two are the ones excluding the clique with four vertices (K4) as a minor, or equivalently, the graphs whose biconnected components are series-parallel. We turn those graphs into a finitely presented free algebra, answering positively a question by Courcelle and Engelfriet, in the case of treewidth two. First we propose a syntax for denoting these graphs: in addition ...

All K4-free graphs with no odd hole and no odd antihole are three-colourable, but what about K4free graphs with no odd hole? They are not necessarily three-colourable, but we prove a conjecture of Ding that they are all four-colourable. This is a consequence of a decomposition theorem for such graphs; we prove that every such graph either has no odd antihole, or belongs to one of two explicitly...

Suppose that n > (log k), where c is a fixed positive constant. We prove that no matter how the edges of Kn are colored with k colors, there is a copy of K4 whose edges receive at most two colors. This improves the previous best bound of k k, where c′ is a fixed positive constant, which follows from results on classical Ramsey numbers.

Let G = (V, E) be a graph where every vertex v ∈ V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v) = {1, . . . , k} for all v ∈ V then a corresponding list coloring is nothing other than an ordinary k-coloring of G. Assume that W ⊆ V is a...