Strong Edge Coloring Research Articles

Strong edge-colouring and induced matchings

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of three edges uses three colours. An induced matching of a graph G is a subset I of edges of G such that the graph induced by the endpoints of I is a matching. In this paper, we prove the NP-completeness of strong 4-, 5-, and 6-edge-colouring and maximum induced matching in some subclasses of subcubic triangle-free planar graphs. We also obtain a tight upper bound for the minimum number of colours in a strong edge-colouring of outerplanar graphs as a function of the maximum degree.

On strong edge-colouring of subcubic graphs

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of length 3 uses three different colours. In this paper we improve some previous results on the strong edge-colouring of subcubic graphs by showing that every subcubic graph with maximum average degree strictly less than 73 (resp. 52, 83, 207) can be strongly edge-coloured with six (resp. seven, eight, nine) colours. These upper bounds are optimal except the one of 83. Also, we prove that every subcubic planar graph without 4-cycles and 5-cycles can be strongly edge-coloured with nine colours.

Strong List Edge Coloring of Subcubic Graphs

We study strong list edge coloring of subcubic graphs, and we prove that every subcubic graph with maximum average degree less than 15/7, 27/11, 13/5, and 36/13 can be strongly list edge colored with six, seven, eight, and nine colors, respectively.

D-strong Edge Colorings of Graphs

For a proper edge coloring of a graph G the palette S(v) of a vertex v is the set of the colors of the incident edges. If S(u) ≠ S(v) then the two vertices u and v of G are distinguished by the coloring. A d-strong edge coloring of G is a proper edge coloring that distinguishes all pairs of vertices u and v with distance 1 ≤ d (u, v) ≤ d. The d-strong chromatic index \({\chi_{d}^{\prime}(G)}\) of G is the minimum number of colors of a d-strong edge coloring of G. Such colorings generalize strong edge colorings and adjacent strong edge colorings as well. We prove some general bounds for \({\chi_{d}^{\prime}(G)}\) , determine \({\chi_{d}^{\prime}(G)}\) completely for paths and give exact values for cycles disproving a general conjecture of Zhang et al. (Acta Math Sinica Chin Ser 49:703–708 2006)).

On the Equitable Adjacent Strong Edge Coloring of K(n,m)

For a graph G(V,E),then is called k-adjacent strong edge coloring of G, is abbreviated k-ASEC, if a k-ASEC of G satisfies then f is called k-equitable adjacent strong edge coloring of G, is called k-EASC for short, and is called the equitable adjacent strong edge chromatic number of G. In this paper, we have proved the equitable adjacent strong edge chromatic number of graph K(n,m) , with n≥ 2,n≡0(mod 2), m≥ 1.

Strong edge-coloring for cubic Halin graphs

A strong edge-coloring of a graph G is a function that assigns to each edge a color such that two edges within distance two apart must receive different colors. The minimum number of colors used in a strong edge-coloring is the strong chromatic index of G. Lih and Liu (2011) [14] proved that the strong chromatic index of a cubic Halin graph, other than two special graphs, is 6 or 7. It remains an open problem to determine which of such graphs have strong chromatic index 6. Our article is devoted to this open problem. In particular, we disprove a conjecture of Shiu et al. (2006) [18] that the strong chromatic index of a cubic Halin graph with characteristic tree a caterpillar of odd leaves is 6.

Bounds and complexity results for strong edge colouring of subcubic graphs

Abstract A strong edge colouring of a graph G is a proper edge colouring such that every path of length 3 uses three colours. In this paper, we give some upper bounds for the minimum number of colours in a strong edge colouring of subcubic graphs as a function of the maximum average degree. We also prove the NP-completeness of the strong edge k-colouring problem for some restricted classes of subcubic planar graphs when k = 4 , 5 , 6 .

On the strong chromatic index of cubic Halin graphs

A strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if they are incident to a common edge or share an endpoint. The strong chromatic index of a graph G , denoted by s χ ′ ( G ) , is the minimum number of colors needed for a strong edge coloring of G . A Halin graph G is a plane graph constructed from a tree without vertices of degree two by connecting all leaves through a cycle. If a cubic Halin graph G is different from two particular graphs N e 2 and N e 4 , then we prove s χ ′ ( G ) ⩽ 7 . This solves a conjecture proposed in W.C. Shiu, W.K. Tam, The strong chromatic index of complete cubic Halin graphs, Appl. Math. Lett. 22 (2009) 754–758.

The strong chromatic index of Halin graphs

A strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if their distance is at most two. The strong chromatic index of a graph G, denoted by sχ′(G), is the minimum number of colors needed for a strong edge coloring of G. A Halin graph G is a plane graph constructed from a tree T without vertices of degree two by connecting all leaves through a cycle C. If a Halin graph G=T∪C is different from a certain necklace Ne2 and any wheel Wn, n≢0(mod3), then we prove that sχ′(G)⩽sχ′(T)+3.

Strong edge colouring of subcubic graphs

A strong edge colouring of a graph G is a proper edge colouring such that every path of length 3 uses three colours. In this paper, we prove that every subcubic graph with maximum average degree strictly less than 15 7 (resp. 27 11 , 13 5 , 36 13 ) can be strong edge coloured with six (resp. seven, eight, nine) colours.

Parity and strong parity edge-colorings of graphs

A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. A parity edge-coloring (respectively, strong parity edge-coloring) is an edge-coloring in which there is no nontrivial parity path (respectively, open parity walk). The parity edge-chromatic number p(G) (respectively, strong parity edge-chromatic number $\widehat{p}(G)$ ) is the least number of colors in a parity edge-coloring (respectively, strong parity edge-coloring) of G. Notice that $\widehat{p}(G) \ge p(G) \ge \chi'(G) \ge \Delta(G)$ for any graph G. In this paper, we determine $\widehat{p}(G)$ and p(G) for some complete bipartite graphs and some products of graphs. For instance, we determine $\widehat{p}(K_{m,n})$ and p(K m,n ) for m?n with n?0,?1,?2 (mod 2?lg?m?).

The adjacent strong edge chromatic number of P<SUB align=right>1 ∨ K<SUB align=right>m,n C<SUB align=right>2 ∨ K<SUB align=right>m,n

An algorithm is designed to search for the k-adjacent strong edge colouring of certain join graphs. Therefore, the adjacent strong edge chromatic number of two join graphs is obtained.

Adjacent strong edge colorings and total colorings of regular graphs

It is conjectured that X as ′ (G) = X t (G) for every k-regular graph G with no C 5 component (k ⩾ 2). This conjecture is shown to be true for many classes of graphs, including: graphs of type 1; 2-regular, 3-regular and (|V(G)| - 2)-regular graphs; bipartite graphs; balanced complete multipartite graphs; k-cubes; and joins of two matchings or cycles.

A Note on Incidence graphs

Abstract An incidence graph of a given graph G, denoted by I(G), has its vertex set V ( I ( G ) ) = { ( v e ) : v ∈ V ( G ) , e ∈ E ( G ) and v is incident to e in G } such that the pair ( u e ) ( v f ) of vertices ( u e ) , ( v f ) ∈ V ( I ( G ) ) is an edge of I(G) if and only if there exists at least one case of u = v , e = f , u v = e or u v = f . Study of Incidence graphs was made by Zhagn Zhong-fu et al. in [Zhagn Zhong-fu, Yao Bing, Li Jing-wen, Linu Lin-zhong, Wang Jian-fang, Xu Bao-gen, On Incidence Graphs, ARS combinatoria 87 (2008), 213-223]. The origin of Incidence graphs can be traced to a paper titled Incidence and Strong edge colorings of graphs by R.A. Brualdi, et al. [Richard A. Brualdi and Jennifer J. Quinn Massey, Incidence and strong edge colorings of graphs, Discrete Mathematics 122 (1993) 51-58]. In this paper, we make a study of domination and related parameters in Incidence graphs.

Invalid proofs on incidence coloring

Incidence coloring of a graph G is a mapping from the set of incidences to a color-set C such that adjacent incidences of G are assigned distinct colors. Since 1993, numerous fruitful results as regards incidence coloring have been proved. However, some of them are incorrect. We remedy the error of the proof in [R.A. Brualdi, J.J.Q. Massey, Incidence and strong edge colorings of graphs, Discrete Math. 122 (1993) 51–58] concerning complete bipartite graphs. Also, we give an example to show that an outerplanar graph with Δ = 4 is not 5-incidence colorable, which contradicts [S.D. Wang, D.L. Chen, S.C. Pang, The incidence coloring number of Halin graphs and outerplanar graphs, Discrete Math. 256 (2002) 397–405], and prove that the incidence chromatic number of the outerplanar graph with Δ ≥ 7 is Δ + 1 . Moreover, we prove that the incidence chromatic number of the cubic Halin graph is 5. Finally, to improve the lower bound of the incidence chromatic number, we give some sufficient conditions for graphs that cannot be ( Δ + 1 ) -incidence colorable.

Θ-graphs of partial cubes and strong edge colorings

Abstract It was conjectured in [5] that the upper bound for the strong chromatic index s ′ ( G ) of bipartite graphs is Δ ( G ) 2 , where Δ ( G ) is the largest degree of vertices in G. In this note we study the strong edge coloring of some classes of bipartite graphs that belong to the class of partial cubes. We introduce the concept of Θ-graph Θ ( G ) of a partial cube G, and show that s ′ ( G ) ⩽ χ ( Θ ( G ) ) for every tree-like partial cube G. As an application of this bound we derive that s ′ ( G ) ⩽ 2 Δ ( G ) if G is a p-expansion graph.

Strong edge-coloring of graphs with maximum degree 4 using 22 colors

In 1985, Erdős and Neśetril conjectured that the strong edge-coloring number of a graph is bounded above by 5 4 Δ 2 when Δ is even and 1 4 ( 5 Δ 2 - 2 Δ + 1 ) when Δ is odd. They gave a simple construction which requires this many colors. The conjecture has been verified for Δ ⩽ 3 . For Δ = 4 , the conjectured bound is 20. Previously, the best known upper bound was 23 due to Horak. In this paper we give an algorithm that uses at most 22 colors.

R-Strong edge colorings of graphs

Let G be a graph and for any natural number r, χ s ′ ( G , r ) denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623–626] it has been proved that for any tree T with at least three vertices, χ s ′ ( T , 1 ) ⩽ Δ ( T ) + 1 . Here we generalize this result and show that χ s ′ ( T , 2 ) ⩽ Δ ( T ) + 1 . Moreover, we show that if for any two vertices u and v with maximum degree d ( u , v ) ⩾ 3 , then χ s ′ ( T , 2 ) = Δ ( T ) . Also for any tree T with Δ ( T ) ⩾ 3 we prove that χ s ′ ( T , 3 ) ⩽ 2 Δ ( T ) - 1 . Finally, it is shown that for any graph G with no isolated edges, χ s ′ ( G , 1 ) ⩽ 3 Δ ( G ) .

Parallel routing algorithms for nonblocking electronic and photonic switching networks

We study the connection capacity of a class of rearrangeable nonblocking (RNB) and strictly nonblocking (SNB) networks with/without crosstalk-free constraint, model their routing problems as weak or strong edge-colorings of bipartite graphs, and propose efficient routing algorithms for these networks using parallel processing techniques. This class of networks includes networks constructed from banyan networks by horizontal concatenation of extra stages and/or vertical stacking of multiple planes. We present a parallel algorithm that runs in O(lg/sup 2/ N) time for the RNB networks of complexities ranging from O(N lg N) to O(N/sup 1.5/ lg N) crosspoints and parallel algorithms that run in O(min{d* lg N, /spl radic/N}) time for the SNB networks of O(N/sup 1.5/ lg N) crosspoints, using a completely connected multiprocessor system of N processing elements. Our algorithms can be translated into algorithms with an O(lg N lg lg N) slowdown factor for the class of N-processor hypercubic networks, whose structures are no more complex than a single plane in the RNB and SNB networks considered.

Strong edge colorings of uniform graphs

For a graph G = ( V ( G ) , E ( G ) ) , a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χ s ( G ) , is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G ( n , p ) was considered in Discrete Math. 281 (2004) 129, Austral. J. Combin. 10 (1994) 97, Austral. J. Combin. 18 (1998) 219 and Combin. Probab. Comput. 11 (1) (2002) 103. In this paper, we consider χ s ( G ) for a related class of graphs G known as uniform or ε -regular graphs. In particular, we prove that for 0 < ε ⪡ d < 1 , all ( d , ε ) -regular bipartite graphs G = ( U ∪ V , E ) with | U | = | V | ⩾ n 0 ( d , ε ) satisfy χ s ( G ) ⩽ ζ ( ε ) Δ ( G ) 2 , where ζ ( ε ) → 0 as ε → 0 (this order of magnitude is easily seen to be best possible). Our main tool in proving this statement is a powerful packing result of Pippenger and Spencer (Combin. Theory Ser. A 51(1) (1989) 24).