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Register a new userNordhaus-Guddam Type Relations of Three Graph Coloring Parameters
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Let G be a simple graph. A coloring of vertices of G is called (i) a 2-proper coloring if vertices at distance 2 receive distinct colors; (ii) an injective coloring if vertices possessing a common neighbor receive distinct colors; (iii) a square coloring if vertices at distance at most 2 receive distinct colors. In this paper, we study inequalities of Nordhaus-Guddam type for the 2-proper chromatic number, the injective chromatic number, and the square chromatic number.
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The fractional matching number of a graph G, is the maximum size of a fractional matching of G. The following sharp lower bounds for a graph G of order n are proved, and all extremal graphs are characterized in this paper. (1)The sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than n/2 , where n is not less than 2. (2) If G and its complement are non-empty, then the sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than (n+1)/2, where n is not less than 28. (3) If G and its complement have no isolated vertices, then the sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than (n+4)/2, where n is not less than 28.
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.
Building upon the notion of Gutman index $operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The emph{Steiner Gutman $k$-index} $operatorname{SGut}_k(G)$ of $G$ is defined by $operatorname{SGut}_k(G)$ $=sum_{Ssubseteq V(G), |S|=k}left(prod_{vin S}deg_G(v)right) d_G(S)$, in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on $operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type results for the parameter $operatorname{SGut}_k$. We obtain sharp upper and lower bounds of $operatorname{SGut}_k(G)+operatorname{SGut}_k(overline{G})$ and $operatorname{SGut}_k(G)cdot operatorname{SGut}_k(overline{G})$ for a connected graph $G$ of order $n$, $m$ edges and maximum degree $Delta$, minimum degree $delta$.
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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