# On the Pseudoachromatic Index of the Complete Graph III

On the Pseudoachromatic Index of the Complete Graph III An edge colouring of a graph G is complete if for any distinct colours $$c_1$$ c 1 and $$c_2$$ c 2 one can find in G adjacent edges coloured with $$c_1$$ c 1 and $$c_2$$ c 2 , respectively. The pseudoachromatic index of G is the maximum number of colours in a complete edge colouring of G. Let $$\psi (n)$$ ψ ( n ) denote the pseudoachromatic index of $$K_n$$ K n . In the paper we proved that if $$x\ge 2$$ x ≥ 2 is an integer and $$n\in \{4x^2-x,\dots ,4x^2+3x-3\}$$ n ∈ { 4 x 2 - x , ⋯ , 4 x 2 + 3 x - 3 } , then $$\psi (n) \le 2x(n-x-1)$$ ψ ( n ) ≤ 2 x ( n - x - 1 ) . Let q be an even integer and let $$m_a=(q+1)^2-a$$ m a = ( q + 1 ) 2 - a . If there is a projective plane of order q, a complete edge colouring of $$K_{m_a}$$ K m a with $$(m_a-a)q$$ ( m a - a ) q colours, $$a\in \{-1,0,\dots ,\frac{q}{2}+1\}$$ a ∈ { - 1 , 0 , ⋯ , q 2 + 1 } , is presented. The main result states that if $$q\ge 4$$ q ≥ 4 is an integer power of 2, then $$\psi (m_a)=(m_a-a)q$$ ψ ( m a ) = ( m a - a ) q for any $$a\in \{-1,0,\dots ,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil -1 \} .$$ a ∈ { - 1 , 0 , ⋯ , 1 + 4 q + 9 2 - 1 } . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Graphs and Combinatorics Springer Journals

# On the Pseudoachromatic Index of the Complete Graph III

, Volume 34 (2) – Jan 22, 2018
11 pages

/lp/springer_journal/on-the-pseudoachromatic-index-of-the-complete-graph-iii-de1YmPUWWp
Publisher
Springer Japan
Subject
Mathematics; Combinatorics; Engineering Design
ISSN
0911-0119
eISSN
1435-5914
D.O.I.
10.1007/s00373-017-1872-6
Publisher site
See Article on Publisher Site

### Abstract

An edge colouring of a graph G is complete if for any distinct colours $$c_1$$ c 1 and $$c_2$$ c 2 one can find in G adjacent edges coloured with $$c_1$$ c 1 and $$c_2$$ c 2 , respectively. The pseudoachromatic index of G is the maximum number of colours in a complete edge colouring of G. Let $$\psi (n)$$ ψ ( n ) denote the pseudoachromatic index of $$K_n$$ K n . In the paper we proved that if $$x\ge 2$$ x ≥ 2 is an integer and $$n\in \{4x^2-x,\dots ,4x^2+3x-3\}$$ n ∈ { 4 x 2 - x , ⋯ , 4 x 2 + 3 x - 3 } , then $$\psi (n) \le 2x(n-x-1)$$ ψ ( n ) ≤ 2 x ( n - x - 1 ) . Let q be an even integer and let $$m_a=(q+1)^2-a$$ m a = ( q + 1 ) 2 - a . If there is a projective plane of order q, a complete edge colouring of $$K_{m_a}$$ K m a with $$(m_a-a)q$$ ( m a - a ) q colours, $$a\in \{-1,0,\dots ,\frac{q}{2}+1\}$$ a ∈ { - 1 , 0 , ⋯ , q 2 + 1 } , is presented. The main result states that if $$q\ge 4$$ q ≥ 4 is an integer power of 2, then $$\psi (m_a)=(m_a-a)q$$ ψ ( m a ) = ( m a - a ) q for any $$a\in \{-1,0,\dots ,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil -1 \} .$$ a ∈ { - 1 , 0 , ⋯ , 1 + 4 q + 9 2 - 1 } .

### Journal

Graphs and CombinatoricsSpringer Journals

Published: Jan 22, 2018

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