Strong chromatic index and Hadwiger number

Partition the vertices of a graph into induced matchings

0. The Hadwiger conjecture [recall that the famous four-color problem is a special case of the Hadwiger conjecture] states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ [where $\chi(G)$ is the chromatic number of $G$, and $\eta(G)$ is the hadwiger number of $G$ (i.e.\ the maximum of $p$ such that $G$ is contractible to the complete graph $K_{p}$)]. In this paper, via an original speech and simple results, we rigorously simplify the understanding of the Hadwiger conjecture. It will appear that to solve the famous Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs.\textbf{1.} The Hadwiger conjecture (see [3] or [4] or [6] or [7] or [8] or [9] or [10]) and the Berge problem (see [2] or [4] or [5] or [6] or [8]) are well known. Recall in a graph $G=[V(G),E(G),\chi(G),\omega(G), \bar{G}]$, $V(G)$ is the set of vertices, $E(G)$ is the set of edges, $\chi(G)$ is the chromatic number, $\omega(G)$ is the clique number and $\bar{G}$ is the complementary graph of $G$. The Hadwiger conjecture [recall that the famous four-color problem is a special case of the Hadwiger conjecture] states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ [where $\eta(G)$ is the hadwiger number of $G$ (i.e.\ the maximum of $p$ such that $G$ is contractible to the complete graph $K_{p}$)]. We say that a graph $B$ is berge if every $B'\in \lbrace B, \bar{B}\rbrace$ does not contain an induced cycle of odd length $\geq 5$. A graph $G$ is perfect if every induced subgraph $G'$ of $G$ satisfies $\chi(G')=\omega(G')$. Indeed, the Berge problem (see [4] or [5] or [6]) consists to show that $\chi(B)=\omega(B)$ for every berge graph $B$ [we recall (see {\bf{[0]), that the Berge problem was solved in a paper of $146$ pages long by Chudnovsky, Robertson, Seymour and Thomas]. In [4], it is presented an original investigation around the Berge problem and the Hadwiger conjecture, and, via two simple Theorems, it is shown that the Berge problem and the Hadwiger conjecture are curiously resembling, so resembling that they seem identical [indeed, they can be restated in ways that resemble each other (see [4])]. Now, in this paper, via only original speech and results, we rigorously simplify the understanding of the Hadwiger conjecture. Moreover, it will appear that to solve the Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs

Strong chromatic index of K4-minor free graphs

Arbitrarily large difference between [formula omitted]-strong chromatic index and its trivial lower bound

List strong edge-coloring of graphs with maximum degree 4

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most 2 receive distinct colors. The strong chromatic index χ′s(G) of a graph G is the minimum number of colors used in a strong edge coloring of G. In an ordering Q of the vertices of G, the back degree of a vertex x of G in Q is the number of vertices adjacent to x, each of which has smaller index than x in Q. Let G be a graph of maximum degree Δ and maximum average degree at most 2k. Yang and Zhu [J. Graph Theory, 83, 334–339 (2016)] presented an algorithm that produces an ordering of the edges of G in which each edge has back degree at most 4kΔ − 2k in the square of the line graph of G, implying that χ′s(G) ≤ 4kΔ − 2k + 1. In this note, we improve the algorithm of Yang and Zhu by introducing a new procedure dealing with local structures. Our algorithm generates an ordering of the edges of G in which each edge has back degree at most (4k − 1)Δ − 2k in the square of the line graph of G, implying that χ′s(G) ≤ (4k − 1)Δ − 2k + 1.

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph Sm(k, l) is a bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, we demonstrate that the conjecture is also valid for a more general family of bipartite graphs. © 1997 John Wiley & Sons, Inc.

Let D = (d1, d2, …, dn) be a graphic sequence with 0 ≤ d1≤ d2≤ ⋯ ≤ dn. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. We say that D is H-free if no graph in R[D] contains H as an induced subgraph. In this paper, we prove that Hadwiger's Conjecture is true for graphs whose degree sequences are claw-free or [Formula: see text]-free.

AVD-total-chromatic number of some families of graphs with [formula omitted

We prove that chie( G) = Delta if Delta ≥ 5 and g ≥ 4, or Delta ≥ 4 and g ≥ 5, or Delta ≥ 3 and g ≥ 9. In addition, if chi(Sigma) > 0, then chie( G) = Delta if Delta ≥ 3 and g ≥ 8 where Delta, g is the maximum degree, the girth of the graph G, respectively.;It is proved that G is not critical if d¯ ≤ 6 and Delta ≥ 8, or d¯ ≤ 203 and Delta ≥ 9. This result generalizes earlier results.;Given a simple plane graph G, an edge-face k-coloring of G is a function &phis; : E(G) ∪ F(G) {lcub}1, ···, k{rcub} such that, for any two adjacent elements a, b ∈ E(G) ∪ F(G), &phis;( a) ≠ &phis;(b). Denote chie( G), chief(G), Delta( G) the edge chromatic number, the edge-face chromatic number and the maximum degree of G, respectively. We prove that chi ef(G) = chie( G) = Delta(G) for any 2-connected simple plane graph G with Delta(G) ≥ 24.

We consider a proper coloring c of edges and vertices in a simple graph and the sum f(v) of colors of all the edges incident to v and the color of a vertex v. We say that a coloring c distinguishes adjacent vertices by sums, if every two adjacent vertices have different values of f. We conjecture that $${\Delta +\,3}$$ Δ + 3 colors suffice to distinguish adjacent vertices in any simple graph. In this paper we show that this holds for complete graphs, cycles, bipartite graphs, cubic graphs and graphs with maximum degree at most three.

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) <TEX>$\rightarrow$</TEX> {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices <TEX>$\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$</TEX> with <TEX>$f(\upsilon_i)$</TEX> = 2 for i = 1, 2, <TEX>$\ldot$</TEX>, k. The weight of a Roman k-dominating function is the value f(V (G)) = <TEX>$\sum_{u{\in}v(G)}$</TEX> f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number <TEX>${\gamma}_{kR}$</TEX>(G) of G. Note that the Roman 1-domination number <TEX>$\gamma_{1R}$</TEX>(G) is the usual Roman domination number <TEX>$\gamma_R$</TEX>(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

Hadwiger's Conjecture is True for Almost Every Graph

Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)\le \Delta(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $|E(G)|>\Delta(G) \lfloor |V(G)|/2 \rfloor$. Since a maximum matching in $G$ can have size at most $\lfloor |V(G)|/2 \rfloor$, it follows that $\chi'(G) = \Delta(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $\Delta$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $\Delta(G) > |V(G)|/3$. In this paper, we show that any $\Delta$-critical graph $G$ is overfull if $\Delta(G) - 7\delta(G)/4\ge (3|V(G)|-17)/4$.

An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is \emph{$H$-free} if $D$ does not contain $H$ as an induced sub(di)graph. The Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest $F$, there is some function $f$ such that every $F$-free graph $G$ with clique number $\omega(G)$ has chromatic number at most $f(\omega(G))$. Aboulker, Charbit, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The \emph{dichromatic number} of a digraph $D$ is the minimum number of colors required to color the vertex set of $D$ so that no directed cycle in $D$ is monochromatic. Aboulker, Charbit, and Naserasr's $\dichi$-boundedness conjecture states that for every oriented forest $F$, there is some function $f$ such that every $F$-free oriented graph $D$ has dichromatic number at most $f(\omega(D))$, where $\omega(D)$ is the size of a maximum clique in the graph underlying $D$. In this paper, we perform the first step towards proving Aboulker, Charbit, and Naserasr's $\dichi$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.