This paper introduces the largest connected subgraph game played on an undirected graph G. In each round, Alice first colours an uncoloured vertex of G red, and then, Bob colours an uncoloured vertex of G blue, with all vertices initially uncoloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph whose order is greater than the order of any blue (red, resp.) connected subgraph. We first prove that, if Alice plays optimally, then Bob can never win, and define a large class of graphs (called reflection graphs) in which the game is a draw. We then show that determining the outcome of the game is PSPACE-complete, even in bipartite graphs of small diameter, and that recognising reflection graphs is GI-hard. We also prove that the game is a draw in paths if and only if the path is of even order or has at least 11 vertices, and that Alice wins in cycles if and only if the cycle is of odd length. Lastly, we give an algorithm to determine the outcome of the game in cographs in linear time.
The 1-2-3 Conjecture states that every nice graph G (without component isomorphic to K2) admits a proper 3-labelling, i.e., a labelling of the edges with 1, 2, 3 such that no two adjacent vertices are incident to the same sum of labels. Another interpretation of this conjecture is that every nice graph G can be turned into a locally irregular multigraph M , i.e., with no two adjacent vertices of the same degree, by replacing each edge by at most three parallel edges. In other words, for every nice graph G, there should exist a locally irregular multigraph M with the same adjacencies and having few edges.We study proper labellings of graphs with the extra requirement that the sum of assigned labels must be as small as possible. That is, given a graph G, we are looking for a locally irregular multigraph M * with the fewest edges possible that can be obtained from G by replacing edges with parallel edges. This problem is quite different from the 1-2-3 Conjecture, as we prove that there is no k such that M * can always be obtained from G by replacing each edge with at most k parallel edges.We investigate several aspects of this problem. We prove that the problem of designing proper labellings with minimum label sum is N Phard in general, but solvable in polynomial time for graphs with bounded treewidth. We also conjecture that every nice connected graph G admits a proper labelling with label sum at most 3 2 |E(G)| + O(1), which we verify for several classes of graphs.
Given a graph G, a k-labelling ℓ of G is an assignment ℓ : E(G) → {1, . . . , k} of labels from {1, . . . , k} to the edges. We say that ℓ is s-proper, m-proper or p-proper, if no two adjacent vertices of G are incident to the same sum, multiset or product, respectively, of labels.Proper labellings are part of the field of distinguishing labellings, and have been receiving quite some attention over the last decades, in particular in the context of the well-known 1-2-3 Conjecture. In recent years, quite some progress was made towards the main questions of the field, with, notably, the analogues of the 1-2-3 Conjecture for m-proper and p-proper labellings being solved. This followed mainly from a better global understanding of these types of labellings.In this note, we focus on a question raised by Paramaguru and Sampathkumar, who asked whether graphs with m-proper 2-labellings always admit s-proper 2-labellings. A negative answer to this question was recently given by Luiz, who provided infinite families of counterexamples. We give a more general result, showing that recognising graphs with m-proper 2-labellings but no s-proper 2-labellings is an NP-hard problem. We also prove a similar result for m-proper 2-labellings and p-proper 2-labellings, and raise a few directions for further work on the topic.
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