Some parameters related to the deficiency of graphs and the deficiency of near-complete graphs

نویسندگان

  • Petros A. Petrosyan
  • Hrant Khachatrian
چکیده

A proper edge-coloring of a graph G is a mapping α : E(G)→ N such that α(e) 6= α(e′) for every pair of adjacent edges e, e′ ∈ E(G). A proper edge-coloring of a graph G is an interval coloring if the colors of edges incident to each vertex of G form an interval of integers. A graph G is interval colorable if it has an interval coloring. Since not all graphs admit an interval coloring, a measure of closeness for a graph to be interval colorable was introduced by Giaro, Kubale and Ma lafiejski in 1999. The deficiency def(G) of a graph G is the minimum number of pendant edges whose attachment to G makes it interval colorable. For a graph G, by wdef (G) and Wdef (G), we denote the minimum and maximum number of colors in proper edge-colorings of G with minimum deficiency, respectively. In this paper, we obtain some bounds on wdef (G) and Wdef (G) for connected graphs. It is known that for the complete graph K2n+1, def(K2n+1) = n (n ∈ N). Recently, Borowiecka-Olszewska, Drgas-Burchardt and Ha luszczak posed the following conjecture on the deficiency of near-complete graphs: if n ∈ N, then def(K2n+1− e) = n− 1. In this paper, we confirm this conjecture.

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عنوان ژورنال:
  • CoRR

دوره abs/1608.00904  شماره 

صفحات  -

تاریخ انتشار 2016