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        1
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        0answers
        38 views

        Coloration of an interval graph with constraints [on hold]

        Given an interval graph that represents a set of tasks, in a given period of time, to be assigned to a set of employees, the objective is to find a minimum coloring of this graph such that the total ...
        0
        votes
        0answers
        34 views

        Algorithmic complexity of deciding the existence of regular $\mathrm{f}$-factors in graphs

        Finding regular $\mathrm{f}$-factors in undirected simple graphs can be reduced to finding a perfect matching by utilizing the gadgets of Tutte or of Lovász and Plummer; there are several algorithms ...
        -2
        votes
        0answers
        45 views

        Polya Enumeration to study proper graph colorings

        Can Polya enumeration technique or its modification be used in evaluating the chromatic polynomial of a simple graph? Specifically, we have to ensure that two distinct points have distinct colors if ...
        4
        votes
        0answers
        199 views

        Reference for results about planar graphs

        A colleague and I are writing a paper in which we need to make use of some basic facts about planar graphs. I would strongly prefer to simply give references for the results if possible, because the ...
        20
        votes
        3answers
        713 views

        When can a graph be oriented to form a Hasse diagram of a finite poset?

        For any finite poset $P=(X,\leq)$ there is an undirected graph $G$ underlying the Hasse diagram of $P$ such that $V(G)=X$ and $E(G)=\{\{u,v\}:u\lessdot v\}$. With that said, is it possible to ...
        8
        votes
        0answers
        235 views
        +50

        Connected subgraphs and their sums

        Let $G=(V,E)$ be an undirected graph with $|V|\geq 4$ such that for any distinct vertices $a_1,a_2,b_1,b_2$, there is a path from $a_1$ to $a_2$ and a (vertex-)disjoint path from $b_1$ to $b_2$. ...
        2
        votes
        1answer
        119 views

        Edge coloring graphs is in P?

        It is known that there exist polynomial time algorithm to approximate the Lovasz number or the supremum of Shannon capacity of graphs. By Vizing's theorem, the graph $G$ has only two chromatic ...
        1
        vote
        1answer
        58 views

        Perfect graphs condition could be weakened?

        The perfect graphs are generally defined as those graphs whose every induced subgraph has its chromatic number equal to its clique number. Now,are there some examples where the clique number of graph ...
        1
        vote
        0answers
        81 views

        Chromatic number of certain graphs with high maximum degree

        Let $G$ be the graph of even order $n$ and size $\ge\frac{n^2}{4}$ which is a Cayley graph on a nilpotent group but not complete. Can the chromatic number of this graph be determined in polynomial ...
        2
        votes
        0answers
        177 views

        Is there a known proof that $R(5,5)\leq 47$ in Ramsey theory?

        As an application to a model describing graphs with partial information, I found what might be an (as yet unverified) proof that $R(5,5)\leq 47$. According to the Dynamic Survey of Ramsey Numbers at ...
        6
        votes
        0answers
        105 views

        Squared squares and partitions of $K_{nn}$

        This is inspired by a recent question. Define a square square sum (SSS) of order $n$ to be any partition $$n^2=\sum_1^tc_ii^2 \tag{*}$$ of $n^2$ into square summands. Call it perfect if all $c_i \leq ...
        15
        votes
        7answers
        972 views

        Examples of proofs by making reduction to a finite set [closed]

        This is a very abstract question, I hope this is appropriate. Suppose $T$ is some claim over some infinite set $A$, for example, let $A$ be the set of all loopless planar graphs, and $T$ be the claim "...
        3
        votes
        1answer
        156 views

        Diameter of Cayley graphs of finite simple groups

        Babai, Kantor and Lubotzky proved in 1989 the following theorem (Sciencedirect link to article). THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ ...
        11
        votes
        4answers
        452 views

        A specific collection of subgraphs in $K_{70, 70}$

        Does there exist a collection of subgraphs $\{\Gamma_i\}_{i = 1}^{24}$ of $K_{70, 70}$, that satisfy the following two properties: 1)$\Gamma_i \cong K_{i, i} \forall 1 \leq i \leq 24$; 2)Any ...
        4
        votes
        0answers
        77 views

        Dinitz Conjecture extension to rectangles

        The Dinitz Conjecture, which was proved later in a more general form by Galvin, stated that given an $n\times n$ array, its elements could be filled exactly like a latin square, where the elements in ...

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