Abstract
In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: $k$-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian. In this coloring, the set $S(v)$ of colors used by edges incident to a vertex $v$ does not intersect $S(u)$ on more than $k$ colors when $u$ and $v$ are adjacent. We provide some sharp upper and lower bounds for $\chi'_{k\text{-int}}$ for several classes of graphs. For $l$-degenerate graphs we prove that $\chi'_{k\text{-int}}(G)\leq (l+1)\Delta -l(k-1)-1$. We improve this bound for subcubic graphs by showing that $\chi'_{2\text{-int}}(G)\leq 6$. We show that calculating $\chi'_{k\text{-int}}(K_n)$ for arbitrary values of $k$ and $n$ is related to some problems in combinatorial set theory and we provide bounds that are tight for infinitely many values of $n$. Furthermore, for complete bipartite graphs we prove that $\chi'_{k\text{-int}}(K_{n,m}) = \left\lceil \frac{mn}{k}\right\rceil$. Finally, we show that computing $\chi'_{k\text{-int}}(G)$ is NP-complete for every $k\geq 1$.An addendum was added to this paper on Jul 4, 2015.
Highlights
A proper edge-coloring of a graph is an assignment of colors to the edges of G such that every pair of adjacent edges receive different colors
For l-degenerate graphs we prove that χk-int(G) (l + 1)∆ − l(k − 1) − 1
We improve this bound for subcubic graphs by showing that χ2-int(G) 6
Summary
A proper edge-coloring of a graph is an assignment of colors to the edges of G such that every pair of adjacent edges receive different colors. An equivalent definition of strong edgecoloring is to say that an edge-coloring φ is a strong edge-coloring if |φ(v) ∩ φ(u)| 1 for each pair u, v of adjacent vertices (the color of uv being the only color in common) This formulation led Muthu, Subramanian and the fifth author [18] to introduce the following relaxed version: a k-intersection edge-coloring of a graph G is a (proper) edge-coloring in which we have |φ(v) ∩ φ(u)| k for each pair u, v of adjacent vertices. The k-intersection chromatic index of G, denoted χk-int(G), is the smallest number of colors in a possible k-intersection edge-coloring of G. After giving few examples, we give general bounds for k-intersection edge-chromatic number in terms of parameters like maximum degree or degeneracy. An l-degenerate graph is a graph any subgraph of which (including the electronic journal of combinatorics 22(2) (2015), #P2.9 itself) has a vertex of degree at most l
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