Total Chromatic Number Research Articles

Edge choosability and total choosability of planar graphs with no 3-cycles adjacent 4-cycles

Two cycles are said to be adjacent if they share a common edge. Let G be a planar graph without triangles adjacent 4-cycles. We prove that χl″(G)≤Δ(G)+2 if Δ(G)≥6, and χl′(G)=Δ(G) and χl″(G)=Δ(G)+1 if Δ(G)≥8, where χl′(G) and χl″(G) denote the list edge chromatic number and list total chromatic number of G, respectively.

Total colorings of planar graphs with maximum degree seven and without intersecting 3-cycles

Let G be a planar graph with maximum degree Δ ≥ 7 and without intersecting 3-cycles; that is, two cycles of length 3 are not incident with a common vertex. Then the total chromatic number of G is Δ + 1 .

Acyclic total colorings of planar graphs without l cycles

A proper total coloring of a graph G such that there are at least 4 colors on those vertices and edges incident with a cycle of G, is called acyclic total coloring. The acyclic total chromatic number of G is the least number of colors in an acyclic total coloring of G. In this paper, it is proved that the acyclic total chromatic number of a planar graph G of maximum degree at least k and without l cycles is at most Δ(G) + 2 if (k, l) ∈ {(6, 3), (7, 4), (6, 5), (7, 6)}.

Total chromatic number of unichord-free graphs

A unichord is an edge that is the unique chord of a cycle in a graph. The class C of unichord-free graphs — that is, graphs that do not contain, as an induced subgraph, a cycle with a unique chord — was recently studied by Trotignon and Vušković (2010) [24], who proved strong structure results for these graphs and used these results to solve the recognition and vertex-colouring problems. Machado et al. (2010) [18] determined the complexity of the edge-colouring problem in the class C and in the subclass C ′ obtained from C by forbidding squares. In the present work, we prove that the total-colouring problem is NP-complete when restricted to graphs in C . For the subclass C ′ , we establish the validity of the Total Colouring Conjecture by proving that every non-complete {square, unichord}-free graph of maximum degree at least 4 is Type 1.

Vertex-Distinguishing Total Coloring of Ladder Graphs (<i>N</i>=0(mod 8) )

A proper total coloring of a simple graph G is called vertex distinguishing if for any two distinct vertices u and v in G, the set of colors assigned to the elements incident to u differs from the set of colors incident to v. The minimal number of colors required for a vertex distinguishing total coloring of G is called the vertex distinguishing total coloring chromatic number. In a paper, we give a “triangle compositor”, by the compositor, we proved that when n=0(mod 8) and , vertex distinguishing total chromatic number of “ladder graphs” is n.

A Note on the Adjacent Vertex Distinguishing Total Chromatic Number of Graph

A total coloring of a simple graph G is called adjacent vertex distinguishing if for any two adjacent and distinct vertices u and v in G, the set of colors assigned to the vertices and the edges incident to u differs from the set of colors assigned to the vertices and the edges incident to v. In this paper we shall prove the series-parallel graph with maximum degree 3 and the series-parallel graph whose the number of edges is the double of maximum degree minus 1 satisfy the adjacent vertex distinguishing total coloring conjecture.

The total-chromatic number of some families of snarks

The total-chromatic number χ T ( G ) is the least number of colours needed to colour the vertices and edges of a graph G such that no incident or adjacent elements (vertices or edges) receive the same colour. It is known that the problem of determining the total-chromatic number is NP-hard, and it remains NP-hard even for cubic bipartite graphs. Snarks are simple connected bridgeless cubic graphs that are not 3-edge-colourable. In this paper, we show that the total-chromatic number is 4 for three infinite families of snarks, namely, the Flower Snarks, the Goldberg Snarks, and the Twisted Goldberg Snarks. This result reinforces the conjecture that all snarks have total-chromatic number 4. Moreover, we give recursive procedures to construct a total-colouring that uses 4 colours in each case.

A decomposition for total‐coloring partial‐grids and list‐total‐coloring outerplanar graphs

AbstractThe total chromatic number χT(G) is the least number of colors sufficient to color the elements (vertices and edges) of a graph G in such a way that no incident or adjacent elements receive the same color. In the present work, we obtain two results on total‐coloring. First, we extend the set of partial‐grids classified with respect to the total‐chromatic number, by proving that every 8‐chordal partial‐grid of maximum degree 3 has total chromatic number 4. Second, we prove a result on list‐total‐coloring biconnected outerplanar graphs. If for each element x of a biconnected outerplanar graph G there exists a set Lx of colors such that |Luw| = max{deg(u) + 1, deg(w) + 1} for each edge uw and |Lv| = 7 − δdeg(v),3 − 2δdeg(v),2 (where δi,j = 1 if i = j and δi,j = 0 if i ≠ j) for each vertex v, then there is a total‐coloring π of graph G such that π(x) ∈ Lx for each element x of G. The technique used in these two results is a decomposition by a cutset of two adjacent vertices, whose properties are discussed in the article. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

Local condition for planar graphs of maximum degree 7 to be 8-totally colorable

The total chromatic number of a graph G , denoted by χ ″ ( G ) , is the minimum number of colors needed to color the vertices and edges of G such that no two adjacent or incident elements get the same color. It is known that if a planar graph G has maximum degree Δ ≥ 9 , then χ ″ ( G ) = Δ + 1 . In this paper, we prove that if G is a planar graph with maximum degree 7 , and for every vertex v , there is an integer k v ∈ { 3 , 4 , 5 , 6 } so that v is not incident with any k v -cycle, then χ ″ ( G ) = 8 .

Fractional total colourings of graphs of high girth

Reed conjectured that for every ϵ > 0 and Δ there exists g such that the fractional total chromatic number of a graph with maximum degree Δ and girth at least g is at most Δ + 1 + ϵ . We prove the conjecture for Δ = 3 and for even Δ ⩾ 4 in the following stronger form: For each of these values of Δ, there exists g such that the fractional total chromatic number of any graph with maximum degree Δ and girth at least g is equal to Δ + 1 .

Generalized total colorings of graphs

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P ,Q)-total coloring ∗Research supported in part by Slovak VEGA Grant 2/0194/10. 210 M. Borowiecki, A. Kemnitz, M. Marangio and P. Mihok of a simple graphG is a coloring of the vertices V (G) and edgesE(G) of G such that for each color i the vertices colored by i induce a subgraph of property P , the edges colored by i induce a subgraph of property Q and incident vertices and edges obtain different colors. In this paper we present some general basic results on (P ,Q)-total colorings. We determine the (P ,Q)-total chromatic number of paths and cycles and, for specific properties, of complete graphs. Moreover, we prove a compactness theorem for (P ,Q)-total colorings.

Total coloring of planar graphs without 6-cycles

A total k - coloring of a graph G is a coloring of V ( G ) ∪ E ( G ) using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ ″ ( G ) is the smallest integer k such that G has a total k -coloring. Let G be a planar graph with maximum degree Δ ( G ) and without 6-cycles. In this paper, it is proved that χ ″ ( G ) = Δ ( G ) + 1 if Δ ( G ) ≥ 5 and G contains no 4-cycles, or Δ ( G ) ≥ 6 and G contains no 5-cycles.

Total chromatic number of {square,unichord}-free graphs

Abstract We determine a surprising class for which edge-colouring is NP-complete but whose graphs are all Type 1. The class of unichord-free graphs was recently studied by Trotignon and Vuskovic [N. Trotignon and K. Vuskovic. A structure theorem for graphs with no cycle with a unique chord and its consequences . To appear in J. Graph Theory. (Available online at http://www.comp.leeds.ac.uk/vuskovi/chord.ps .)] in the context of vertex-colouring. Machado, Figueiredo and Vuskovic [R. C. S. Machado, C. M. H. de Figueiredo, and K. Vuskovic. Chromatic index of graphs with no cycle with unique chord . In: Proc. of the ALIO/EUROWorkshop on Applied Optimization (2008). Full paper submitted to Theoret. Comput. Sci.] established the NP-completeness of edge-colouring unichord-free graphs. For the subclass of {square,unichord}-free graphs, an interesting complexity dicothomy holds [R. C. S. Machado, C. M. H. de Figueiredo, and K. Vuskovic. Chromatic index of graphs with no cycle with unique chord . In: Proc. of the ALIO/EUROWorkshop on Applied Optimization (2008). Full paper submitted to Theoret. Comput. Sci.]: if the maximum degree is 3, the edge-colouring is NP-complete, otherwise, the problem is polynomial. Subsequently, Machado and Figueiredo [R. C. S. Machado and C. M. H. de Figueiredo. Total chromatic number of chordless graphs . In: Proc. of the Cologne-Twente Workshop (2009). Full paper submitted to Discrete Appl. Math.] settled the validity of the Total-Colouring Conjecture for {square,unichord}-free graphs by proving that non-complete {square,unichord}-free graphs of maximum degree at least 4 are Type 1. In the present work, we prove that non-complete {square,unichord}-free graphs of maximum degree 3 are Type 1, establishing the polynomiality of total-colouring restricted to {square,unichord}-free graphs.

Total coloring of embedded graphs of maximum degree at least ten

A total k-coloring of a graph G is a coloring of V(G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ″(G) is the smallest integer k such that G has a total k-coloring. In this paper, it is proved that the total chromatic number of any graph G embedded in a surface Σ of Euler characteristic χ(Σ) ⩾ 0 is Δ(G) + 1 if Δ(G) ⩾ 10, where Δ(G) denotes the maximum degree of G.

Total coloring of planar graphs of maximum degree eight

The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ ″ ( G ) . It is known that if a planar graph G has maximum degree Δ ⩾ 9 , then χ ″ ( G ) = Δ + 1 . Recently Hou et al. (Graphs and Combinatorics 24 (2008) 91–100) proved that if G is a planar graph with maximum degree 8 and with either no 5-cycles or no 6-cycles, then χ ″ ( G ) = 9 . In this Note, we strengthen this result and prove that if G is a planar graph with maximum degree 8, and for each vertex x, there is an integer k x ∈ { 3 , 4 , 5 , 6 , 7 , 8 } such that there is no k x -cycle which contains x, then χ ″ ( G ) = 9 .

The Local Nature of List Colorings for Graphs of High Girth

We consider list coloring problems for graphs $\mathcal{G}$ of girth larger than $c\log_{\Delta-1}n$, where n and $\Delta\geq3$ are, respectively, the order and the maximum degree of $\mathcal{G}$, and c is a suitable constant. First, we determine that the edge and total list chromatic numbers of these graphs are $\chi'_l(\mathcal{G})=\Delta$ and $\chi”_l(\mathcal{G})=\Delta+1$. This proves that the general conjectures of Bollobás and Harris [Graphs Combin., 1 (1985), pp. 115–127], Behzad [The total chromatic number, in Combinatorial Mathematics and Its Applications (Proc. Conf., Oxford, 1969), Academic Press, London, 1971, pp. 1–8], Vizing [Diskret. Analiz., 3 (1964), pp. 25–30], and Juvan, Mohar, and Škrekovski [Combin. Probab. Comput., 7 (1998), pp. 181–188] hold for this particular class of graphs. Moreover, our proofs exhibit a certain degree of “locality,” which we exploit to obtain an efficient distributed algorithm able to compute both kinds of optimal list colorings. Also, using an argument similar to one of Erdös, we show that our algorithm can compute k-list vertex colorings of graphs having girth larger than $c\log_{k-1}n$.

The Last Fraction of a Fractional Conjecture

Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most $\Delta+1+\varepsilon$. The conjecture was proven to be true when $\Delta=3$ or $\Delta$ is even. We settle the conjecture by proving it for the remaining cases.

A Characterization of Graphs with Fractional Total Chromatic Number Equal to

Abstract For a simple graph of maximum degree Δ, the complexity of computing the fractional total chromatic number is unknown. Trivially it is at least Δ + 1 . Kilakos and Reed proved that it is at most Δ + 2 . In this paper, we strengthen this by characterizing exactly those simple graphs with fractional total chromatic number Δ + 2 . This yields a simple linear-time algorithm to determine whether a given graph has fractional chromatic number Δ + 2 .

On total chromatic number of direct product graphs

Coloring of the graph products, especially vertex and edge coloring, has been widely researched for all types of graph products. For total graph coloring, as combination of edge and vertex coloring, Behzad and Vizing set Total Coloring Conjecture in mid 1960s. In this paper, we prove the conjecture for two specific direct graph products, for direct product of path and arbitrary graph G, Pn×G, where χ′(G)=Δ(G), and expand the proof onto direct product of arbitrary cycle and a path Pn, Cm×Pn. At the same time, the proofs provide the algorithms to color such graphs.

Total colorings and list total colorings of planar graphs without intersecting 4-cycles

Suppose that G is a planar graph with maximum degree Δ and without intersecting 4-cycles, that is, no two cycles of length 4 have a common vertex. Let χ ″ ( G ) , χ l ′ ( G ) and χ l ″ ( G ) denote the total chromatic number, list edge chromatic number and list total chromatic number of G , respectively. In this paper, it is proved that χ ″ ( G ) = Δ + 1 if Δ ≥ 7 , and χ l ′ ( G ) = Δ and χ l ″ ( G ) = Δ + 1 if Δ ( G ) ≥ 8 . Furthermore, if G is a graph embedded in a surface of nonnegative characteristic, then our results also hold.