By moving to millimeter wave (mmWave) frequencies, base stations (BSs) will be densely deployed provide seamless coverage in sixth generation (6G) mobile communication systems, which, unfortunately, leads severe cell-edge problem. In addition, with massive multiple-input-multiple-output (MIMO) antenna arrays employed at BSs, the beamspace channel is sparse for each user, and thus there no need ...

We prove that every claw-free graph G that does not contain a clique on ∆(G) ≥ 9 vertices can be ∆(G)− 1 colored.

In 1988, Chvátal and Sbihi [4] proved a decomposition theorem for claw-free perfect graphs. They showed that claw-free perfect graphs either have a clique-cutset or come from two basic classes of graphs called elementary and peculiar graphs. In 1999, Maffray and Reed [6] successfully described how elementary graphs can be built using line-graphs of bipartite graphs using local augmentation. How...

If G is a claw-free graph, then there is a graph cl(G) such that (i) G is a spanning subgraph of cl(G), (ii) cl(G) is a line graph of a triangle-free graph, and (iii) the length of a longest cycle in G and in cl(G) is the same. A suucient condition for hamiltonicity in claw-free graphs, the equivalence of some conjectures on hamiltonicity in 2-tough graphs and the hamiltonicity of 7-connected c...

A connected n-chromatic graph G is double-critical if for all the edges xy of G, the graph G−x−y is (n−2)-chromatic. In 1966, Erdős and Lovász conjectured that the only double-critical n-chromatic graph is Kn. This conjecture remains unresolved for n ≥ 6. In this short note, we verify this conjecture for claw-free graphs G of chromatic number 6.

Ryjacek [Z. Ryjáček: On a closure concept in claw-free graphs. Journal of Combinatorial Theory Ser. B 70 (1997), 217-224] defined a powerful closure operation cl(G) on claw-free graphs G. Very recently, Ryjacek, Yoshimoto and the talker developed the closure operation cl (G) on claw-free graphs which preserves the (non)-existence of a 2-factor. In this talk, we introduce a closure operation cl(...

If C is a subclass of the class of claw-free graphs, then C is said to be stable if, for any G ∈ C, the local completion of G at any vertex is also in C. If cl is a closure operation that turns a claw-free graph into a line graph by a series of local completions and C is stable, then cl(G) ∈ C for any G ∈ C. In the talk we consider stability of hereditary classes of claw-free graphs defined in ...

A graph is prismatic if for every triangle T , every vertex not in T has exactly one neighbour in T . In this paper and the next in this series, we prove a structure theorem describing all prismatic graphs. This breaks into two cases depending whether the graph is 3-colourable or not, and in this paper we handle the 3-colourable case. (Indeed we handle a slight generalization of being 3-coloura...