We resolve the computational complexity of Graph IsomorphIsm for classes of graphs characterized by two forbidden induced subgraphs H 1 and H 2 for all but six pairs (H 1 , H 2). Schweitzer had previously shown that the number of open cases was finite, but without specifying the open cases. Grohe and Schweitzer proved that Graph IsomorphIsm is polynomial-time solvable on graph classes of bounded cliquewidth. Our work combines known results such as these with new results. By exploiting a relationship between Graph IsomorphIsm and clique-width, we simultaneously reduce the number of open cases for boundedness of clique-width for (H 1 , H 2)-free graphs to five.

A signed graph (G, σ) is a graph G along with a function σ : E(G) → {+, −}. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks. The signed chromatic number of a signed graph (G, σ) is the minimum number of vertices |V (H)| of a signed graph (H, π) to which (G, σ) admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention in the last decades, especially due to their strong connections to the theories of graph coloring and graph minors. These homomorphisms have been particularly studied through the scope of the signed chromatic number. In this work, we provide new results and bounds on the signed chromatic number of several families of signed graphs (planar graphs, triangle-free planar graphs, K n -minor-free graphs, and bounded-degree graphs).

Can every connected graph burn in ⌈ √ n⌉ steps? While this conjecture remains open, we prove that it is asymptotically true when the graph is much larger than its growth, which is the maximal distance of a vertex to a well-chosen path in the graph. In fact, we prove that the conjecture for graphs of bounded growth boils down to a finite number of cases. Through an improved (but still weaker) bound for all trees, we argue that the conjecture almost holds for all graphs with minimum degree at least 3 and holds for all large enough graphs with minimum degree at least 4. The previous best lower bound was 23.How fast can a rumor propagate in a graph? One measure of that, introduced by Bonato, Janssen and Roshanbin [BJR16], is the burning number b(G) of a graph G. At step 1, we set a vertex on fire. At every step i 2, all neighbours of a vertex on fire catch fire themselves, and we set a new vertex on fire. If at the end of step k the whole graph is on fire, then the graph is k-burnable. The burning number b(G) of G is defined to be the least k such that G is k-burnable.A graph with n isolated vertices is trivially not (n − 1)-burnable. We focus on connected graphs. Paths are an interesting special case. For a path P n on n vertices, it is not hard to check that b(P n ) = ⌈ √ n⌉. When introducing the notion, Bonato et al.[BJR16] conjectured that paths are, essentially, the worst case for the burning number of a graph. Conjecture 1.1 (Bonato et al. [BJR16]). Every connected graph G satisfies b(G) ⌈ |V (G)|⌉.Conjecture 1.1 is only known to hold with a constant factor. For any connected graph G and any spanning tree T of G, we have b(G) b(T ). Therefore, it is sufficient to prove that the conjecture holds for trees. While Conjecture 1.1 is still open in general, it has been shown to hold for specific graph classes. Kamali, Miller and Zhang showed [KMZ20] that the conjecture holds for graphs of minimum degree at least 23. Here, we observe that the conjecture almost holds for all connected graphs of minimum degree at least 3, and fully holds for those of minimum degree at least 4 that are large enough.

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