7,816 research outputs found
Game chromatic number of Cartesian and corona product graphs
The game chromatic number is investigated for Cartesian product and corona product of two graphs and . The exact values for the game chromatic number of Cartesian product graph of is found, where is a star graph of order . This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let be the path graph on vertices and be the cycle graph on vertices. We have determined the exact values for the game chromatic number of corona product graphs and
Hedetniemi's conjecture for Kneser hypergraphs
One of the most famous conjecture in graph theory is Hedetniemi's conjecture
stating that the chromatic number of the categorical product of graphs is the
minimum of their chromatic numbers. Using a suitable extension of the
definition of the categorical product, Zhu proposed in 1992 a similar
conjecture for hypergraphs. We prove that Zhu's conjecture is true for the
usual Kneser hypergraphs of same rank. It provides to the best of our knowledge
the first non-trivial and explicit family of hypergraphs with rank larger than
two satisfying this conjecture (the rank two case being Hedetniemi's
conjecture). We actually prove a more general result providing a lower bound on
the chromatic number of the categorical product of any Kneser hypergraphs as
soon as they all have same rank. We derive from it new families of graphs
satisfying Hedetniemi's conjecture. The proof of the lower bound relies on the
-Tucker lemma
On the set chromatic number of the join and comb product of graphs
A vertex coloring c : V(G) → of a non-trivial connected graph G is called a set coloring if NC(u) ≠NC(v) for any pair of adjacent vertices u and v. Here, NC(x) denotes the set of colors assigned to vertices adjacent to x. The set chromatic number of G, denoted by χs (G), is defined as the fewest number of colors needed to construct a set coloring of G. In this paper, we study the set chromatic number in relation to two graph operations: join and comb prdocut. We determine the set chromatic number of wheels and the join of a bipartite graph and a cycle, the join of two cycles, the join of a complete graph and a bipartite graph, and the join of two bipartite graphs. Moreover, we determine the set chromatic number of the comb product of a complete graph with paths, cycles, and large star graphs
Graph Product Structure for h-Framed Graphs
Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph classes for which the product structure holds, such as to planar graphs [Dujmovi? et al., J. ACM, 67(4), 22:1-38, 2020].
In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the h-framed graphs, a graph class that includes 1-planar, optimal 2-planar, and k-map graphs (for appropriate values of h). We establish a graph product structure theorem for h-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most 3, and of a clique of size 3? h/2 ?+? h/3 ?-1. This allows us to improve over the previous structural theorems for 1-planar and k-map graphs. Our results constitute significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and p-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of 1-planar graphs and k-map graphs from 115 to 82 and from ? 33/2(k+3 ? k/2? -3)? to ? 33/2 (3? k/2 ?+? k/3 ?-1) ?, respectively. We also employ the product structure machinery to improve the current upper bounds on the twin-width of 1-planar graphs from O(1) to 80. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions
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