scholarly journals Snarks with total chromatic number 5

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Gunnar Brinkmann ◽  
Myriam Preissmann ◽  
Diana Sasaki
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Total Coloring ◽  
Computer Search ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Total Chromatic Number ◽  
Vertex Number ◽  
Chordless Cycle

Graph Theory International audience A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by χT(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with χT = 4 are said to be Type 1, and cubic graphs with χT = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n≥40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open.

scholarly journals Total Coloring Conjecture for Certain Classes of Graphs

Algorithms ◽  
2018 ◽  
Vol 11 (10) ◽  
pp. 161 ◽  
Author(s):  
R. Vignesh ◽  
J. Geetha ◽  
K. Somasundaram
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Product Line ◽  
Total Coloring ◽  
Lexicographic Product ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Total Coloring Conjecture

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

Total colorings of circulant graphs

2020 ◽  
pp. 2150050
Author(s):  
J. Geetha ◽  
K. Somasundaram ◽  
Hung-Lin Fu
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Total Coloring ◽  
Circulant Graphs ◽  
Total Chromatic Number ◽  
Number Formula ◽  
Total Coloring Conjecture

The total chromatic number [Formula: see text] is the least number of colors needed to color the vertices and edges of a graph [Formula: see text] such that no incident or adjacent elements (vertices or edges) receive the same color. Behzad and Vizing proposed a well-known total coloring conjecture (TCC): [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. For the powers of cycles, Campos and de Mello proposed the following conjecture: Let [Formula: see text] denote the graphs of powers of cycles of order [Formula: see text] and length [Formula: see text] with [Formula: see text]. Then, [Formula: see text] In this paper, we prove the Campos and de Mello’s conjecture for some classes of powers of cycles. Also, we prove the TCC for complement of powers of cycles.

The Smarandachely Adjacent Vertex Distinguishing E-Total Coloring of some Join Graphs

2013 ◽  
Vol 475-476 ◽  
pp. 379-382
Author(s):  
Mu Chun Li ◽  
Shuang Li Wang ◽  
Li Li Wang
Keyword(s):  
Chromatic Number ◽  
Total Coloring ◽  
Adjacent Vertex ◽  
Analysis Method ◽  
Total Chromatic Number ◽  
Join Graph ◽  
Total Coloring Conjecture

Using the analysis method and the function of constructing the Smarandachely adjacent vertex distinguishing E-total coloring function, the Smarandachely adjacent vertex distinguishing E-total coloring of join graphs are mainly discussed, and the Smarandachely adjacent vertex distinguishing E-total chromatic number of join graph are obtained. The Smarandachely adjacent vertex distinguishing E-total coloring conjecture is further validated.

scholarly journals On the equitable total chromatic number of cubic graphs

2016 ◽  
Vol 209 ◽  
pp. 84-91 ◽  
Author(s):  
S. Dantas ◽  
C.M.H. de Figueiredo ◽  
G. Mazzuoccolo ◽  
M. Preissmann ◽  
V.F. dos Santos ◽  
...  
Keyword(s):  
Chromatic Number ◽  
Cubic Graphs ◽  
Total Chromatic Number

Splitting on classes of helm graphs and its equitabletotal chromatic number

2020 ◽  
pp. 635-642
Author(s):  
J. Veninstine Vivik ◽  
D. Dafik
Keyword(s):  
Chromatic Number ◽  
Total Coloring ◽  
Total Chromatic Number ◽  
Splitting Graph

The equitable total coloring of a graph $G$ is the different colors used to color all the vertices and edges of $G$, in the order that adjacent vertices and edges are assigned with least different $k$-colors and can be partitioned into colors sets which differ by maximum one. The minimum of $k$-colors required is known as the equitable total chromatic number. In this paper the splitting graph of Helm and Closed Helm graph is constructed and its equitable total chromatic number is acquired.

2-distance vertex-distinguishing total coloring of graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850018
Author(s):  
Yafang Hu ◽  
Weifan Wang
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Number Formula ◽  
Proper Total Coloring ◽  
Distance Formula

A [Formula: see text]-distance vertex-distinguishing total coloring of a graph [Formula: see text] is a proper total coloring of [Formula: see text] such that any pair of vertices at distance [Formula: see text] have distinct sets of colors. The [Formula: see text]-distance vertex-distinguishing total chromatic number [Formula: see text] of [Formula: see text] is the minimum number of colors needed for a [Formula: see text]-distance vertex-distinguishing total coloring of [Formula: see text]. In this paper, we determine the [Formula: see text]-distance vertex-distinguishing total chromatic number of some graphs such as paths, cycles, wheels, trees, unicycle graphs, [Formula: see text], and [Formula: see text]. We conjecture that every simple graph [Formula: see text] with maximum degree [Formula: see text] satisfies [Formula: see text].

scholarly journals Some Results on the Structure of Multipoles in the Study of Snarks

10.37236/3629 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
M. A. Fiol ◽  
J. Vilaltella
Keyword(s):  
Lower Bound ◽  
Edge Coloring ◽  
The State ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimum Order ◽  
Exact Number ◽  
Empty Subset ◽  
Free Edges

Multipoles are the pieces we obtain by cutting some edges of a cubic graph in one or more points. As a result of the cut, a multipole $M$ has vertices attached to a dangling edge with one free end, and isolated edges with two free ends. We refer to such free ends as semiedges, and to isolated edges as free edges. Every 3-edge-coloring of a multipole induces a coloring or state of its semiedges, which satisfies the Parity Lemma. Multipoles have been extensively used in the study of snarks, that is, cubic graphs which are not 3-edge-colorable. Some results on the states and structure of the so-called color complete and color closed multipoles are presented. In particular, we give lower and upper linear bounds on the minimum order of a color complete multipole, and compute its exact number of states. Given two multipoles $M_1$ and $M_2$ with the same number of semiedges, we say that $M_1$ is reducible to $M_2$ if the state set of $M_2$ is a non-empty subset of the state set of $M_1$ and $M_2$ has less vertices than $M_1$. The function $v(m)$ is defined as the maximum number of vertices of an irreducible multipole with $m$ semiedges. The exact values of  $v(m)$ are only known for $m\le 5$. We prove that tree and cycle multipoles are irreducible and, as a byproduct, that $v(m)$ has a linear lower bound.

scholarly journals Super local edge anti-magic total coloring of paths and its derivation

2020 ◽  
Vol 3 (2) ◽  
pp. 126
Author(s):  
Fawwaz Fakhrurrozi Hadiputra ◽  
Denny Riama Silaban ◽  
Tita Khalis Maryati
Keyword(s):  
Chromatic Number ◽  
Minimum Weight ◽  
Edge Coloring ◽  
Simple Graph ◽  
Total Coloring ◽  
Local Edge

<p>Suppose <em>G</em>(<em>V,E</em>) be a connected simple graph and suppose <em>u,v,x</em> be vertices of graph <em>G</em>. A bijection <em>f</em> : <em>V</em> ∪ <em>E</em> → {1,2,3,...,|<em>V</em> (<em>G</em>)| + |<em>E</em>(<em>G</em>)|} is called super local edge antimagic total labeling if for any adjacent edges <em>uv</em> and <em>vx</em>, <em>w</em>(<em>uv</em>) 6= <em>w</em>(<em>vx</em>), which <em>w</em>(<em>uv</em>) = <em>f</em>(<em>u</em>)+<em>f</em>(<em>uv</em>)+<em>f</em>(<em>v</em>) for every vertex <em>u,v,x</em> in <em>G</em>, and <em>f</em>(<em>u</em>) &lt; <em>f</em>(<em>e</em>) for every vertex <em>u</em> and edge <em>e</em> ∈ <em>E</em>(<em>G</em>). Let γ(<em>G</em>) is the chromatic number of edge coloring of a graph <em>G</em>. By giving <em>G</em> a labeling of <em>f</em>, we denotes the minimum weight of edges needed in <em>G</em> as γ<em>leat</em>(<em>G</em>). If every labels for vertices is smaller than its edges, then it is be considered γ<em>sleat</em>(<em>G</em>). In this study, we proved the γ sleat of paths and its derivation.</p>

scholarly journals General Vertex-Distinguishing Total Coloring of Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Chanjuan Liu ◽  
Enqiang Zhu
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Total Coloring ◽  
Total Chromatic Number ◽  
Minimum Integer

The general vertex-distinguishing total chromatic number of a graphGis the minimum integerk, for which the vertices and edges ofGare colored usingkcolors such that any two vertices have distinct sets of colors of them and their incident edges. In this paper, we figure out the exact value of this chromatic number of some special graphs and propose a conjecture on the upper bound of this chromatic number.

Total coloring conjecture for vertex, edge and neighborhood corona products of graphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950014
Author(s):  
Radhakrishnan Vignesh ◽  
Jayabalan Geetha ◽  
Kanagasabapathi Somasundaram
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Tight Bound ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Total Coloring Conjecture

A total coloring of a graph [Formula: see text] is an assignment of colors to the elements of the graph [Formula: see text] such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any simple graph [Formula: see text], [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. In this paper, we prove the tight bound of the total coloring conjecture for the three types of corona products (vertex, edge and neighborhood) of graphs.

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