823 research outputs found

    Dynamic Chromatic Number of Regular Graphs

    Full text link
    A dynamic coloring of a graph GG is a proper coloring such that for every vertex v∈V(G)v\in V(G) of degree at least 2, the neighbors of vv receive at least 2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}. PhD thesis, West Virginia University, 2001.] that if GG is a kk-regular graph, then Ο‡2(G)βˆ’Ο‡(G)≀2\chi_2(G)-\chi(G)\leq 2. In this paper, we prove that if GG is a kk-regular graph with Ο‡(G)β‰₯4\chi(G)\geq 4, then Ο‡2(G)≀χ(G)+Ξ±(G2)\chi_2(G)\leq \chi(G)+\alpha(G^2). It confirms the conjecture for all regular graph GG with diameter at most 2 and Ο‡(G)β‰₯4\chi(G)\geq 4. In fact, it shows that Ο‡2(G)βˆ’Ο‡(G)≀1\chi_2(G)-\chi(G)\leq 1 provided that GG has diameter at most 2 and Ο‡(G)β‰₯4\chi(G)\geq 4. Moreover, we show that for any kk-regular graph GG, Ο‡2(G)βˆ’Ο‡(G)≀6ln⁑k+2\chi_2(G)-\chi(G)\leq 6\ln k+2. Also, we show that for any nn there exists a regular graph GG whose chromatic number is nn and Ο‡2(G)βˆ’Ο‡(G)β‰₯1\chi_2(G)-\chi(G)\geq 1. This result gives a negative answer to a conjecture of [A. Ahadi, S. Akbari, A. Dehghan, and M. Ghanbari. \newblock On the difference between chromatic number and dynamic chromatic number of graphs. \newblock {\em Discrete Math.}, In press].Comment: 8 page

    Recognizing Graph Theoretic Properties with Polynomial Ideals

    Get PDF
    Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Groebner bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
    • …
    corecore