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Recent results on well-balanced orientations

Algorithms & optimization · math.CO · math.OC · posed by Attila Bernáth, Satoru Iwata, Tamás Király, Zoltán Király, Zoltán Szigeti · open

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Statement

For every graph G=(V+s,E)G = (V + s, E) with d(s)≥4d(s) \ge 4 there exist rs,st∈Ers, st \in E such that for every best-balanced orientation G⃗rt\vec{G}_{rt} of GrtG_{rt}, G⃗:=G⃗rt−rt+rs+st\vec{G} := \vec{G}_{rt} - rt + rs + st is a best-balanced orientation of GG.

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  • Recent results on well-balanced orientations
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Reconstructed statement: in finite loopless multigraphs, an orientation is best-balanced if it is smooth,

    ∣d+(v)−d−(v)∣≤1∀v,|d^+(v)-d^-(v)|\le 1\quad\forall v,

    and well-balanced,

    λG⃗(x,y)≥⌊λG(x,y)/2⌋∀x≠y,\lambda_{\vec G}(x,y)\ge \big\lfloor \lambda_G(x,y)/2\big\rfloor \quad\forall x\ne y,

    where λ\lambda denotes local edge/arc-connectivity. For edges rs,strs,st, GrtG_{rt} is obtained by deleting them and adding rtrt; lifting replaces r→tr\to t by r→s→tr\to s\to t. This is the standard Nash-Williams “best-balanced” terminology used in the cited splitting-off section.

    Result: The statement is false.

    Let

    V(G)={s,a,b,c,d,e,f}V(G)=\{s,a,b,c,d,e,f\}

    and

    E(G)={sa,sb,sc,sf,ac,ad,ae,af,be,bf,ce}.E(G)=\{sa,sb,sc,sf,ac,ad,ae,af,be,bf,ce\}.

    Then dG(s)=4d_G(s)=4. Also λG(s,a)=4\lambda_G(s,a)=4: the four edge-disjoint ss-aa paths are

    sa,sbea,sca,sfa,sa,\quad sbea,\quad sca,\quad sfa,

    and dG(s)=4d_G(s)=4 gives the matching upper bound.

    For every ordered split of two edges incident with ss, the following table gives a best-balanced orientation of the split graph whose lift is not well-balanced. In the table, x→yx\to y is the oriented new split edge, and repeated arcs mean parallel edges in the split graph.

    new arcorientation of Gxybad cut after liftinga→bs→c,f→s,a→c,a→d,e→a,f→a,b→e,b→f,c→e,a→ba→s, X={a,c,d,e}b→as→c,f→s,a→c,a→d,a→e,f→a,e→b,b→f,c→e,b→aa→s, X={a,c,d,e}a→cs→b,f→s,a→c,d→a,a→e,f→a,e→b,b→f,c→e,a→cs→a, X={s,b,c,e}c→as→b,f→s,a→c,a→d,e→a,f→a,e→b,b→f,c→e,c→aa→s, X={a,d}a→fs→b,c→s,a→c,a→d,e→a,f→a,e→b,b→f,c→e,a→fs→a, X={s,b,f}f→as→b,c→s,a→c,a→d,a→e,f→a,b→e,b→f,e→c,f→aa→s, X={a,c,d,e}b→cs→a,f→s,a→c,a→d,a→e,f→a,e→b,b→f,c→e,b→ca→s, X={a,c,d,e}c→bs→a,f→s,a→c,a→d,a→e,f→a,b→e,b→f,e→c,c→ba→s, X={a,c,d,e}b→fs→a,c→s,a→c,a→d,e→a,f→a,e→b,b→f,c→e,b→fa→s, X={a,d}f→bs→a,c→s,a→c,a→d,e→a,f→a,e→b,b→f,c→e,f→ba→s, X={a,d}c→fs→a,b→s,a→c,a→d,a→e,f→a,b→e,f→b,e→c,c→fa→s, X={a,c,d,e}f→cs→a,b→s,a→c,a→d,a→e,f→a,e→b,b→f,c→e,f→ca→s, X={a,c,d,e}\begin{array}{c|l|l} \text{new arc} & \text{orientation of }G_{xy} & \text{bad cut after lifting}\\ \hline a\to b & s\to c,f\to s,a\to c,a\to d,e\to a,f\to a,b\to e,b\to f,c\to e,a\to b & a\to s,\ X=\{a,c,d,e\}\\ b\to a & s\to c,f\to s,a\to c,a\to d,a\to e,f\to a,e\to b,b\to f,c\to e,b\to a & a\to s,\ X=\{a,c,d,e\}\\ a\to c & s\to b,f\to s,a\to c,d\to a,a\to e,f\to a,e\to b,b\to f,c\to e,a\to c & s\to a,\ X=\{s,b,c,e\}\\ c\to a & s\to b,f\to s,a\to c,a\to d,e\to a,f\to a,e\to b,b\to f,c\to e,c\to a & a\to s,\ X=\{a,d\}\\ a\to f & s\to b,c\to s,a\to c,a\to d,e\to a,f\to a,e\to b,b\to f,c\to e,a\to f & s\to a,\ X=\{s,b,f\}\\ f\to a & s\to b,c\to s,a\to c,a\to d,a\to e,f\to a,b\to e,b\to f,e\to c,f\to a & a\to s,\ X=\{a,c,d,e\}\\ b\to c & s\to a,f\to s,a\to c,a\to d,a\to e,f\to a,e\to b,b\to f,c\to e,b\to c & a\to s,\ X=\{a,c,d,e\}\\ c\to b & s\to a,f\to s,a\to c,a\to d,a\to e,f\to a,b\to e,b\to f,e\to c,c\to b & a\to s,\ X=\{a,c,d,e\}\\ b\to f & s\to a,c\to s,a\to c,a\to d,e\to a,f\to a,e\to b,b\to f,c\to e,b\to f & a\to s,\ X=\{a,d\}\\ f\to b & s\to a,c\to s,a\to c,a\to d,e\to a,f\to a,e\to b,b\to f,c\to e,f\to b & a\to s,\ X=\{a,d\}\\ c\to f & s\to a,b\to s,a\to c,a\to d,a\to e,f\to a,b\to e,f\to b,e\to c,c\to f & a\to s,\ X=\{a,c,d,e\}\\ f\to c & s\to a,b\to s,a\to c,a\to d,a\to e,f\to a,e\to b,b\to f,c\to e,f\to c & a\to s,\ X=\{a,c,d,e\} \end{array}

    Each listed split orientation is smooth by direct degree count. In every split graph, all local connectivities are at most 33, except pairs involving the pendant vertex dd, whose requirement is 00. The displayed orientations contain a directed closed walk through {s,a,b,c,e,f}\{s,a,b,c,e,f\}, hence are strongly connected on these six vertices, so they are well-balanced.

    After lifting, the bad cut XX contains the source but not the target and has exactly one outgoing arc. Thus the lifted orientation has directed local connectivity at most 11 between ss and aa in one direction, while ⌊λG(s,a)/2⌋=2\lfloor\lambda_G(s,a)/2\rfloor=2. Hence the lift is not best-balanced.

    Since all twelve ordered choices of two incident edges at ss fail, no pair rs,strs,st satisfies the proposed property.

    Citation: Original open problem: Bernáth, Iwata, T. Király, Z. Király, and Szigeti, “Recent results on well-balanced orientations,” Discrete Optimization 5 (2008), 663–676, Section 6, Question 5. The counterexample above is not cited there.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE2

    PASS

    The proposed counterexample attacks the stated universal splitting-off question. The graph has d(s)=4d(s)=4 and λG(s,a)=4\lambda_G(s,a)=4. For each ordered split at ss, the table gives a smooth orientation of the split graph; since all relevant undirected local connectivities in the split graph are at most 33 and the six non-pendant vertices are strongly connected, these orientations are best-balanced. After lifting, the listed cut separates ss and aa in one direction with only one outgoing arc, so the lifted orientation has directed local connectivity at most 1<⌊4/2⌋=21<\lfloor 4/2\rfloor=2. Thus every possible pair of incident edges at ss fails the required “for every best-balanced orientation” property. I found no evidence of an already-known counterexample in the available searches.

    Novelty assessment

    TYPE2

    Classification rationale: Genuinely new as far as I can determine. It gives a small counterexample to an explicit splitting-off question from Bernáth–Iwata–T. Király–Z. Király–Szigeti. The result is narrow and proof-by-case/checking rather than methodologically deep, so it is low-end TYPE2, not TYPE3. Still, because the question was a proposed route to a simpler proof of Nash-Williams’ best-balanced orientation theorem, a clean counterexample is plausibly publishable as a short note in a standard graph theory/combinatorial optimization journal.

    Literature check: I found no prior counterexample or stronger published disproof. Searches over OpenAlex for “best-balanced”, “splitting off”, “for every best-balanced orientation”, and exact fragments of Question 5 returned only the original paper, related theses/surveys, and later work on well-balanced-orientation complexity/odd-vertex pairings. The 2022 Hörsch–Szigeti paper still frames Nash-Williams’ best-balanced theorem as lacking a simple proof and does not mention a resolution of this splitting-off question. The cited works and later citations of the original paper do not appear to contain the proposed counterexample.

    Citation: Original open problem: A. Bernáth, S. Iwata, T. Király, Z. Király, and Z. Szigeti, “Recent results on well-balanced orientations,” Discrete Optimization 5 (2008), 663–676, Section 6, Question 5, DOI 10.1016/j.disopt.2008.03.001.

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