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We study choosability with separation which is a constrained version of list coloring of graphs. A (k, d)-list assignment L of a graph G is a function that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) andThis concept is also known as choosability with separation. We prove that planar graphs without 4-cycles are (3, 1)-choosable and that planar graphs without 5-cycles and 6-cycles are (3, 1)-choosable. In addition, we give an alternative and slightly stronger proof that triangle-free planar graphs are (3, 1)-choosable.

A graph is (d1, . . . , d k )-colorable if its vertex set can be partitioned into k nonempty subsets so that the subgraph induced by the ith part has maximum degree at most di for each i ∈ {1, . . . , k}. It is known that for each pair (d1, d2), there exists a planar graph with girth 4 that is not (d1, d2)-colorable. This sparked the interest in finding the pairs (d1, d2) such that planar graphs with girth at least 5 are (d1, d2)-colorable. Given d1 ≤ d2, it is known that planar graphs with girth at least 5 are (d1, d2)-colorable if either d1 ≥ 2 and d1 + d2 ≥ 8 or d1 = 1 and d2 ≥ 10. We improve an aforementioned result by providing the first pair (d1, d2) in the literature satisfying d1 + d2 ≤ 7 where planar graphs with girth at least 5 are (d1, d2)-colorable. Namely, we prove that planar graphs with girth at least 5 are (3, 4)-colorable.

A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter ρ, every graph G with sufficiently large ρ(G) contains a "well-structured" induced subgraph H with large ρ(H). The classical Ramsey's theorem deals with the case when the graph parameter under consideration is the number of vertices; there is also a Ramsey-type theorem regarding connected graphs.Given a graph G, the matching number and the induced matching number of G is the maximum size of a matching and an induced matching, respectively, of G. In this paper, we formulate Ramsey-type theorems for the matching number and the induced matching number regarding connected graphs. Along the way, we obtain a Ramsey-type theorem for the independence number regarding connected graphs as well.

A graph G is (d1,…,dk)‐colorable if its vertex set can be partitioned into k sets V1,-0.28em…,Vk, such that for each i∈{1,…,k}, the subgraph of G induced by Vi has maximum degree at most di. The Four Color Theorem states that every planar graph is (0,0,0,0)‐colorable, and a classical result of Cowen, Cowen, and Woodall shows that every planar graph is (2,2,2)‐colorable. In this paper, we extend both of these results to graphs on surfaces. Namely, we show that every graph embeddable on a surface of Euler genus g>0 is (0,0,0,9g−4)‐colorable and (2,2,9g−4)‐colorable. Moreover, these graphs are also (0,0,O(g),O(g))‐colorable and (2,O(g),O(g))‐colorable. We also prove that every triangle‐free graph that is embeddable on a surface of Euler genus g is (0,0,O(g))‐colorable. This is an extension of Grötzsch's Theorem, which states that triangle‐free planar graphs are (0,0,0)‐colorable. Finally, we prove that every graph of girth at least 7 that is embeddable on a surface of Euler genus g is (0,O(g))‐colorable. All these results are best possible in several ways as the girth condition is sharp, the constant maximum degrees cannot be improved, and the bounds on the maximum degrees depending on g are tight up to a constant multiplicative factor.

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