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Kaleidoscopic Colorings of Graphs

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Chartrand G., English S., Zhang P.

Discussiones Mathematicae Graph Theory

2017

tom 37

nr 3

711-727

For an r-regular graph G, let c : E(G) → [k] = {1, 2, . . . , k}, k ≥ 3, be an edge coloring of G, where every vertex of G is incident with at least one edge of each color. For a vertex v of G, the multiset-color cm(v) of v is defined as the ordered k-tuple (a1, a2, . . . , ak) or a1a2 … ak, where ai (1 ≤ i ≤ k) is the number of edges in G colored i that are incident with v. The edge coloring c is called k-kaleidoscopic if cm(u) ≠ cm(v) for every two distinct vertices u and v of G. A regular graph G is called a k-kaleidoscope if G has a k-kaleidoscopic coloring. It is shown that for each integer k ≥ 3, the complete graph Kk+3 is a k-kaleidoscope and the complete graph Kn is a 3-kaleidoscope for each integer n ≥ 6. The largest order of an r-regular 3-kaleidoscope is [...] (r−12) $\left( {\matrix{{r - 1} \cr 2 } } \right)$ . It is shown that for each integer r ≥ 5 such that r ≢ 3 (mod 4), there exists an r-regular 3-kaleidoscope of order [...] (r−12) $\left( {{{r - 1} \over 2}} \right)$ .

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Chartrand G.

1 English S.

1 Zhang P.

1 2017

Wyniki wyszukiwania

Kaleidoscopic Colorings of Graphs