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Hadwiger Conjecture has been an open problem for over a half century1,6, which says that there is at most a complete graph Kt but no Kt+1 for every t-colorable graph. A few cases of Hadwiger Conjecture, such as 1, 2, 3, 4, 5, 6-colorable graphs have been completely proved to convince all1-5, but the proofs are tremendously difficult for over the 5-colorable graph6,7. Although the development of graph theory inspires scientists to understand graph coloring deeply, it is still an open problem for over 7-colorable graphs6,7. Therefore, we put forward a brand new chromatic graph configuration and show how to describe the graph coloring issues in chromatic space. Based on this idea, we define a chromatic plane and configure the chromatic coordinates in Euler space. Also, we find a method to prove Hadwiger Conjecture for every 8-coloring graph feasible.
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We propose an algorithm to reduce a k-chromatic graph to a complete graph of largest possible order through a well defined sequence of contractions. We introduce a new matrix called transparency matrix and state its properties. We then define correct contraction procedure to be executed to get largest possible complete graph from given connected graph. We finally give a characterization for k-chromatic graphs and use it to settle Hadwigers conjecture.
We prove that every partial function with finite domain and range can be effectively simulated through sequential colorings of graphs. Namely, we show that given a finite set $S={0,1,ldots,m-1}$ and a number $n geq max{m,3}$, any partial function $varphi:S^{^p} to S^{^q}$ (i.e. it may not be defined on some elements of its domain $S^{^p}$) can be effectively (i.e. in polynomial time) transformed to a simple graph $matr{G}_{_{varphi,n}}$ along with three sets of specified vertices $$X = {x_{_{0}},x_{_{1}},ldots,x_{_{p-1}}}, Y = {y_{_{0}},y_{_{1}},ldots,y_{_{q-1}}}, R = {Kv{0},Kv{1},ldots,Kv{n-1}},$$ such that any assignment $sigma_{_{0}}: X cup R to {0,1,ldots,n-1} $ with $sigma_{_{0}}(Kv{i})=i$ for all $0 leq i < n$, is {it uniquely} and {it effectively} extendable to a proper $n$-coloring $sigma$ of $matr{G}_{_{varphi,n}}$ for which we have $$varphi(sigma(x_{_{0}}),sigma(x_{_{1}}),ldots,sigma(x_{_{p-1}}))=(sigma(y_{_{0}}),sigma(y_{_{1}}),ldots,sigma(y_{_{q-1}})),$$ unless $(sigma(x_{_{0}}),sigma(x_{_{1}}),ldots,sigma(x_{_{p-1}}))$ is not in the domain of $varphi$ (in which case $sigma_{_{0}}$ has no extension to a proper $n$-coloring of $matr{G}_{_{varphi,n}}$).
We study a combinatorial coloring game between two players, Spoiler and Algorithm, who alternate turns. First, Spoiler places a new token at a vertex in $G$, and Algorithm responds by assigning a color to the new token. Algorithm must ensure that tokens on the same or adjacent vertices receive distinct colors. Spoiler must ensure that the token graph (in which two tokens are adjacent if and only if their distance in $G$ is at most $1$) has chromatic number at most $w$. Algorithm wants to minimize the number of colors used, and Spoiler wants to force as many colors as possible. Let $f(w,G)$ be the minimum number of colors needed in an optimal Algorithm strategy. A graph $G$ is online-perfect if $f(w,G) = w$. We give a forbidden induced subgraph characterization of the class of online-perfect graphs. When $G$ is not online-perfect, determining $f(w,G)$ seems challenging; we establish $f(w,G)$ asymptotically for some (but not all) of the minimal graphs that are not online-perfect. The game is motivated by a natural online coloring problem on the real line which remains open.
Motivated by the ErdH{o}s-Faber-Lovasz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We discuss several conjectures for list edge coloring linear hypergraphs that generalize both EFL and Vizings theorem for graphs. For example, we conjecture that in a linear hypergraph of rank 3, the list edge chromatic number is at most 2 times the maximum degree plus 1. We show that for sufficiently large fixed rank and sufficiently large degree, the conjectures are true.
Hadwigers conjecture is one of the most important and long-standing conjectures in graph theory. Reed and Seymour showed in 2004 that Hadwigers conjecture is true for line graphs. We investigate this conjecture on the closely related class of total graphs. The total graph of $G$, denoted by $T(G)$, is defined on the vertex set $V(G)sqcup E(G)$ with $c_1,c_2in V(G)sqcup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to or incident on each other in $G$. We first show that there exists a constant $C$ such that, if the connectivity of $G$ is at least $C$, then Hadwigers conjecture is true for $T(G)$. The total chromatic number $chi(G)$ of a graph $G$ is defined to be equal to the chromatic number of its total graph. That is, $chi(G)=chi(T(G))$. Another well-known conjecture in graph theory, the total coloring conjecture or TCC, states that for every graph $G$, $chi(G)leqDelta(G)+2$, where $Delta(G)$ is the maximum degree of $G$. We show that if a weaker version of the total coloring conjecture (weak TCC) namely, $chi(G)leqDelta(G)+3$, is true for a class of graphs $mathcal{F}$ that is closed under the operation of taking subgraphs, then Hadwigers conjecture is true for the class of total graphs of graphs in $mathcal{F}$. This motivated us to look for classes of graphs that satisfy weak TCC. It may be noted that a complete proof of TCC for even 4-colorable graphs (in fact even for planar graphs) has remained elusive even after decades of effort; but weak TCC can be proved easily for 4-colorable graphs. We noticed that in spite of the interest in studying $chi(G)$ in terms of $chi(G)$ right from the initial days, weak TCC is not proven to be true for $k$-colorable graphs even for $k=5$. In the second half of the paper, we make a contribution to the literature on total coloring by proving that $chi(G)leqDelta(G)+3$ for every 5-colorable graph $G$.
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